Proceedings of the Summer School on “Representation theory of Algebras, Finite and Reductive Groups” “Babeş-Bolyai” University Cluj, 1998; pp. 127-155.

A COURSE ON INTEGRAL GROUP RINGS

WOLFGANG KIMMERLE

1. The Problems

Denote by ℤG the integral group ring of a finite group G. The structure of the unit group U(ℤG) has been of fundamental interest throughout the last sixty years. It may be considered as a topic in non-commutative number theory, in integral representation theory of finite groups, in ring theory, and last not least in group theory. The whole story began 1939 with G. Higman’s thesis and this was indeed motivated by questions arising from algebraic topology and algebraic K - theory. Until 1997 it was an open question whether the torsion subgroups of U(ℤG) are determined by G. In particular the following famous question has been at the centre of most investigations.

The Isomorphism Problem Does a ring isomorphism between ℤG and ℤH imply that G and H are isomorphic as groups ?

1997 M.Hertweck [17] constructed two non - isomorphic groups of order 2^{21} · 97^{28} with isomorphic integral group rings. This negative answer is not solely a counterexample. Its construction and its nature fits very well into concepts developed in the last ten years fits and which nowadays gives some understanding of the torsion subgroups of the group of units of an integral group ring. 1971 E. Dade constructed two non - isomorphic metabelian groups such that their group algebras over all fields are isomorphic [11]. But note that for metabelian groups the isomorphism problem has a positive answer [64], see also 3.6.

For many classes of finite groups the answer is positive and this happens in a very strong sense. The following conjectures of H.Zassenhaus date around 1976 and these

128 WOLFGANG KIMMERLE

conjectures, if true for a group G and its integral group ring ℤG supply such a strong positive answer. They have been stated for the integral group ring ℤG. But they make sense and they have been studied in the slightly more general situation when ℤ is replaced by a G - adapted ring S ^{1} of characteristic zero. If S is G - adapted and of characteristic zero then we call SG freely also an integral group ring.

The Zassenhaus Conjectures [58] Let G be a finite group and let S be G - adapted. Suppose that S has characteristic zero and denote the quotient field of S by K. Denote by V (SG) the normal subgroup of the group units U(SG) which consists of the units of augmentation 1, cf. 3.1.

1. (ZC 1) Let u be a unit of finite order of V (SG). Then u is conjugate within KG to an element of G.

2. (ZC 2) Let U be a subgroup of V (SG) with the same order as G. Then U is conjugate within KG to G.

3. (ZC 3) Let U be a finite subgroup of V (SG). Then U is conjugate within KG to a subgroup of G. As background we mention the following fundamental facts about torsion subgroups

of integral group rings. Under the assumption that SG is an arbitrary integral group ring they are due to Saksonov, cf. section 4. [54]. Let SG be an integral group ring.

• A finite subgroup of V (SG) consists of elements which are linearly independent over S. Its order divides the order of G.

• If H is an arbitrary finite group, then SG and SH are isomorphic if, and only if, H has the same order as G and H is isomorphic to a subgroup of V (SG). In particular a finite subgroup H of V (SG) of the same order as G is an S - basis of SG. ^{2} Moreover, since an arbitrary group isomorphism between isomorphic group bases

H extends uniquely to a S - algebra automorphism of SG, this permits the study of conjecture (ZC 2) for many groups G by considerations of normalized ^{3} S - algebra automorphisms. More precisely by the Noether - Skolem theorem [10, (3.62)] (ZC 2) is valid provided (IP) has an affirmative answer and the following property (AUT) holds.

1Let G be a finite group. An integral domain S is called G - adapted if no prime dividing the order of G is invertible in S.

2Such subgroups are usually called a group basis of SG.

3An S - algebra automorphism is called normalized if it preserves the augmentation.

A COURSE ON INTEGRAL GROUP RINGS 129

(AUT) Let σ be a normalized S - algebra automorphism of SG. Then σ admits a Zassenhaus decomposition, i.e. σ may be written as the composition of an S -algebra automorphism induced from a group automorphism α of G and a central automorphism τ.

If (ZC 3) is true then the torsion part of V (SG) is determined by G modulo conjugation within KG. Clearly it is a natural question to study conjugation within SG itself. This leads to the so - called conjugacy problem

(CP) Let H be a finite subgroup of SG. Is H conjugate to a subgroup of G within SG ?

Since (ZC 2) is not valid for all finite groups, see section 5.2, but holds for many classes it is worthwile to look after a certain replacement of it. This can be done under the side order that the replacement still implies a positive answer to (IP) [24] or not and leads to the following variations.

(ZC 2)_{π} Let SG be an integral group ring. Let H be a group basis of SG and let π be a set of rational primes. Then there exists a ring automorphism σ_{π} such that σ_{π}(G) = H and σ_{π} restricted to the class sums of p-elements for each p ∈ π is the identity.

Of particular interest is of course the case when π consists of a single prime p. In this case we write simply (ZC 2)_{p}.

Let SG be an integral group ring and let H be a group basis of SG. Then

(Syl)_{1} p - subgroups of G and H are conjugate within KG.
(Syl)_{2} If U ≤ V (SG) and U is a p - group. Then U is

conjugate to a subgroup of G. Clearly (Syl)_{2} implies (Syl)_{1} and (ZC)_{p} is equivalent to (Syl)_{1} and (IP).

2. Integral versus other representations

It will be assumed that the reader is familiar with ordinary representation theory and also with the basic facts about modular representations. The first aim of this section is to give an example which clearly demonstrates special features of integral representations. We shall see that even the Jordan - Hoelder theorem is not present for ℤG - lattices. Throughout all sections G denotes a finite group.

Definition 2.1. Let R be a commutative ring. A group homomorphism ϕ : G ⟶ GL(n, R) is called R - representation of G of degree n. ϕ is called

130 WOLFGANG KIMMERLE

ordinary if R = ℂ or more generally

if R is a field of characteristic zero.

rational if R = ℚ
modular if R is a field of characteristic p and p||G|.
integral if R = ℤ or more generally

if R is integral with respect to G .

Example 2.2. • The R - representations of degree 1 are determined by the torsion

units of U(R) and by \overline{G}. and are given by Hom(\overline{G}, U(R)).

• Permutation representations. Let H ≤ G, n = |G : H| and let RG/H be the free R - module on the set of left cosets G/H. Left multiplication with g ∈ G on G/H induces an R - linear endomorphism of RG/H denoted by μ(g). The map μ : G ⟶ GL(n, R) is an R - representation and is called the permutation representation of G with respect to H.

• Let G = S_{3} be the symmetric group of degree 3. Consider the following matrices of GL(2, ℤ).

A_{1} := ( \begin{matrix}−2 & −3 \\ 1 & 1 \end{matrix} ) B_{1} := ( \begin{matrix}−2 & −3 \\ 1 & 2 \end{matrix} )

A_{2} := ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) B_{2} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

Then there are unique representations ϕ_{1} : S_{3} ⟶ GL(2, ℤ) and ϕ_{2} : S_{3} ⟶ GL(2, ℤ) defined by ϕ_{i} : (123) ↦ A_{i} and ϕ_{i} : (12) ↦ B_{i} .

Definition 2.3. The following notions are as usual. Let ϕ, ϕ_{1}, ϕ_{2} be R - representations. Then

• ϕ is called faithful if Kerϕ = 1.

• ϕ_{1} and ϕ_{2} are called equivalent, written ϕ_{1} ∼ ϕ_{2} , if exists T ∈ GL(n, R) such that

T^{−1}ϕ_{1}(g)T = ϕ_{2}(g) ∀g ∈ G

• ϕ is called reducible if in block matrix notation

ϕ ∼ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) .

Otherwise ϕ is called irreducible.

Example 2.4. a) Let G be the dihedral group of order 8 with the usual presentation

G =< a, b; a^{4}, b^{2}, a^{b} = a^{−1} > .

Consider the matrices

A_{1} := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) B_{1} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

B_{2} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

The representations ϕ_{1} defined by a ↦ A_{1}, b ↦ B_{1} and ϕ_{2} defined by a ↦ A_{1}, b ↦ B_{2} are as ℤ - representations irreducible and not equivalent. Using ordinary character theory one sees that they are as ℚ - representations however equivalent.

b) Let G, A_{1}, A_{2} and B_{2} be as in part a). Let in block matrix notation

C_{1} := ( \begin{matrix}A_{1} & 0 \\ 0 & A_{1} \end{matrix} ) D_{1} := ( \begin{matrix}B_{1} & B_{2} \\ 0 & B_{1} \end{matrix} )

and

D_{2} := ( \begin{matrix}B_{2} & B_{1} \\ 0 & B_{2} \end{matrix} )

Then the representations ψ_{1} defined by a ↦ C_{1}, b ↦ D_{1} and ψ_{2} defined by a ↦ C_{1}, b ↦ D_{2} are equivalent ℤ - representations.

Definition 2.5. • A finitely generated RG - module M is called RG - lattice if M

is as R - module projective.

• Let ϕ be an R - representation of degree n of G. Then ϕ(G) acts on the free R - module M = R^{n} as usual by matrix multiplication. Thus ϕ yields naturally an RG - lattice which will be denoted by M_{ϕ}. Note that M_{ϕ_{1}} ≅ M_{ϕ_{2}} if and only if ϕ_{1} ∼ ϕ_{2}.

• A ring R is called hereditary if submodules of finitely generated projective R - modules are projective. Examples for such rings are given by Dedekind rings, in particular the ring of algebraic integers in an algebraic number field is hereditary. Note that if R is hereditary then RG - submodules of RG - lattices are RG - lattices.

• Let N be an RG - lattice. Then a sublattice M of N is called pure, if N/M is an RG - lattice.

• Assume that R is a principal ideal domain. Then RG - lattices are as R - modules free and have an R - dimension. Moreover the number of equivalence classes of R - representations of degree n coincides with the number of isomorphism classes of RG - lattices of dimension n. An RG - lattice M is called reducible, if it has a non - trivial submodule of smaller R - dimension. Otherwise M is called irreducible.

From now on we consider mainly the case when R = ℤ.

132 WOLFGANG KIMMERLE

Lemma 2.6. Let M be a ℤG − lattice.

a) Let N ≤ M. Then there exists a pure sublattice Ñ of M such that Ñ contains N and dim_{ℤ}Ñ = dim_{ℤ}N.

b) There exists a chain of sublattices

0 = T_{0} < T_{1} < … < T_{k} = M

such that T_{i}/T_{i−1} is an irreducible ℤG - lattice.

Looking at part b) of Lemma 2.6 it is a natural question to ask whether these irreducible ℤG - lattices are in the sense of a Jordan - Hoelder theorem unique.

Example 2.7. There is no Jordan - Hoelder theorem for ℤG - lattices. A counterexample is provided by the representations ψ_{i} of Examples 2.4 . One easily computes that M_{ψ_{1}} ≅ M_{ψ_{2}}. Obviously there is a chain

0 = T_{0} < T_{1} < T_{2} = M_{ψ_{i}}

with T_{1}/T_{0} ≅ T_{2}/T_{1} ≅ M_{ϕ_{i}}. But M_{ϕ_{1}} is not isomorphic to M_{ϕ_{2}} because the corresponding ℤ - representations are not equivalent.

2.8. Denote by ℤG − mod the category of finitely generated ℤG - modules and by ℤG− mod ^{0} the category of ℤG - lattices. The following questions should be regarded as fundamental questions on integral representations.

• For which groups G is ℤG of finite representation type, i.e. ℤG− mod ^{0} has only finitely many isomorphism classes of indecomposable ℤG - lattices.

• What are the projective indecomposable ℤG - lattices.

• Is the Krull - Schmidt - Azumaya theorem (cf. I, 4.6), abbreviated in the sequel as KSA - theorem, valid in ℤG − mod ^{0}? For which groups G is the KSA - theorem valid in ℤG − mod ^{0}?

The first two questions are answered by the following well known theorems.

Theorem 2.9 (A. Jones). [10, (33.6)], [22] ℤG has finite representation type if and only if for each p ∈ π(G) the Sylow p - subgroups of G are cyclic of order p or p^{2}.

Note that F_{p}G is of finite representation type if and only if the Sylow p - subgroups of G are cyclic. So modularly there is no restriction on the order of the Sylow subgroups.

Theorem 2.10 (R. Swan). [10, §32], [60] Let M be a finitely generated projective ℤG - module.

a) Then for each prime p the ℤ_{p}G - module M_{p} = ℤ_{p} ⊗_{ℤ} M is free.

b) M ≅ (ℤG)^{r} ⊕ J, where J is a projective ideal of ℤG.

In order to get answers to the third question we consider permutation lattices.

Lemma 2.11. The permutation module ℤG/H is indecomposable.

Proof. Assume that ℤG/H = B ⊕ C is a non-trivial decomposition. Denote by β the projection from ℤG/H onto B. Then β^{2} = β and 0 ≠ β ≠ 1 in End_{ℤG}(ℤG/H). Let G/H = {H, g_{2}H, …, g_{k}H} and

β(H) = a_{11}H + a_{21}g_{2}H + … + a_{k1}g_{k}H.

Then the trace of β with respect to the ℤ - basis G/H is k · a_{11}. On the other hand if one calculates the trace of β with respect to a ℤ - basis through B and C it follows that

0 < trace(β) = dim_{ℤ} B < k .

But a_{11} ∈ ℤ gives the desired contradiction. q.e.d.

Note that we established - H = 1 - without Theorem 2.10 that ℤG is indecomposable.

2.12. The notion of genus. Let N, M be ℤG - lattices. Write N ∨ M, if N_{p} ≅ M_{p} for all p. Then the set Γ_{M} = {N; N ∨ M} is called the genus of M.

2.13. Semi - localization. Let π be a finite set of primes. The ring ℤ_{π} is semi-local, i.e. it has only finitely many maximal ideals. If π ⊃ π(G) then ℤ_{π} is G - adapted. The basic properties of lattices over group rings with semi - local coefficients are.

a) For each N ∈ ℤ_{π}G − mod ^{0} exists M ∈ ℤG − mod with N ≅ M_{π}, where M_{π} = ℤ_{π} ⊗_{ℤ} M.

b) Let M, N ∈ ℤG − mod ^{0}. Then M_{π} ≅ N_{π} if and only if M_{p} ≅ N_{p} ∀p ∈ π.

c) M ∨ N ⇔ M_{p} ≅ N_{p} ∀p ∈ π(G) ⇔ M_{π(G)} ≅ N_{π(G)}.

d) (Semi - local cancellation) Let π be a finite set of primes and A, B, C ∈ ℤ_{π}G −

mod ^{0}. Then

A ⊕ C ≅ B ⊕ C ⇒ A ≅ B

Proof. (of part d)) By c) it suffices to prove the statement when π consists of a single prime p. Clearly

A ⊕ C ≅ B ⊕ C ⇒ A/p^{l}A ⊕ C/p^{l}C ≅ B/p^{l}B ⊕ C/p^{l}C.

In ℤ/p^{l}ℤG − mod the KSA - theorem holds. Hence we get that A/p^{l}A ≅ B/p^{l}B. Apply now Marandas lemma:

Let |G| = p^{k} · n with (k, n) = 1. Then ℤ_{p}G - lattices M, N are isomorphic if and only if M/p^{k+1}M ≅ N/p^{k+1}N.

With l ≥ k + 1 we get now that A ≅ B and d) is proved. q.e.d.

134 WOLFGANG KIMMERLE

Remark 2.14. By 2.12 c) and 2.10 a) it follows that in ℤ_{π(G)}G − mod projective modules are free. Moreover the same holds if ℤ_{π(G)} is replaced by a semi-local Dedekind domain in which no prime divisor of |G| is invertible.

2.15. Let π = π(G). Let F : ℤ_{π}G − mod ⟶ ℤ_{π}H − mod be an equivalence of categories. Then F (ℤ_{π}G) is certainly projective and indecomposable. By 2.14 it follows that F (ℤ_{π}G) ≅ ℤ_{π}H Hence

(F (End_{ℤ_{π}G}(ℤ_{π}(G)), ∘) ≅ (End(ℤ_{π}H), ∘) ≅ End(ℤ_{π}G, ∘).

Thus

U(ℤ_{π}G) ≅ Aut_{ℤ_{π}G}(ℤ_{π}G) ≅ Aut_{ℤ_{π}G}(ℤ_{π}G) ≅ U(ℤ_{π}H).

We shall see later, cf. 4.3, that this implies that ℤ_{π}G ≅ ℤ_{π}H.

Definition 2.16. Let P, Q be finitely generated ℤG - projectives. Define an equivalence relation by

P ∼ Q ⇔ P ⊕ (ℤG)^{r} ≅ Q ⊕ (ℤG)^{s}

for some r, s ∈ ℕ_{0}. Denote the equivalence class of P by [P] and let Cl(ℤG) be the set of equivalence classes. Then Cl(ℤG) with the addition [P] + [Q] := [P ⊕ Q] is called the class group of ℤG.

It is an easy exercise that Cl(ℤG) is an abelian group.

Theorem 2.17 (Jordan - Zassenhaus). [10, §24] There are only finitely many isomorphism classes of ℤG - lattices of a given rank.

Corollary 2.18. Cl(ℤG) is finite.

Proof. By 2.10 b) the elements of Cl(ℤG) are given by [0] and [J], where J is a non - free projective ideal of ℤG. By 2.10 a) all such ideals J have ℤ - rank |G|. Hence the result follows by Theorem 2.17. q.e.d.

2.19. Results on class groups [10, §50].

a) (Rim) Let p be a prime. Cl(ℤC_{p}) ≅ Cl(ℤ[ζ].), where ζ denotes a primitive p - th root of unity. Note that Cl(ℤ[ζ] > 0, if p ≥ 23.

b) (Cassou - Nogues) If G is abelian then Cl(ℤG) = 0 if and only if G is cyclic of order ≤ 11 or G is cyclic of order 13, 14, 17, 19 or G = C_{2} × C_{2}.

c) (Endo - Hironaka) If G is non - abelian and Cl(ℤG)) = 0, then G ≅ A_{4}, A_{5}, S_{4} or D_{n}.

We come now to the third question.

Theorem 2.20. Assume that for ℤG − mod ^{0} the KSA - theorem is valid. Then G ≅ C_{p}, where p is a prime ≤ 19 or G ≅ D_{m} and m is a power of 2.

Proof. Step 1. G has to be a p - group. Let T_{1}, T_{2}, … T_{k} be Sylow p_{i} - subgroups of G with |G| = |T_{1}| · . . . · |T_{k}|. Claim. ℤ is a direct summand of ℤG/T_{1} ⊕ … ⊕ ℤG/T_{k}. Note that by Lemma 2.11 the permutation modules ℤG/T_{i} are indecomposable and if G is not a p - group then no ℤG/T_{i} is isomorphic to ℤ. Thus the KSA - theorem fails and Step 1 is established.

Proof of the claim. Consider the ℤG - epimorphisms ε_{i} : ℤG/T_{i} ⟶ ℤ defined analogously to the augmentation map by ∑ z_{l}g_{k}T_{i} ↦ ∑ z_{l}. This yields

μ : ⊕_{i=1}^{k} ℤG/T_{i} ⟶ ℤ

defined by

(x_{1}, …, x_{k}) ↦ ∑_{i=1}^{k} ε_{i}(x_{i}).

ℤG/T_{i} has a unique submodule on which G acts trivially given by

(ℤG/T_{i})^{G} = _{ℤ} < t_{i} := T_{i} + g_{2}T_{i} + … + g_{l_{i}}T_{i} >,

where l_{i} = |G : T_{i}| denotes the index of T_{i} in G. Note that ε_{i}(t_{i}) = l_{i} and that gcd(l_{1}, …, l_{k}) = 1. Thus exist α_{i} such that 1 = ∑ α_{i}l_{i}. Define ν : ℤ ⟶ ⊕ℤG/T_{i} by 1 ↦ (α_{1}t_{1}, …, α_{k}t_{k}) then ν splits μ and the claim follows.

Step 2. If G̃ is an image of G then the KSA - theorem is valid for ℤG̃ − mod ^{0} provided it holds for ℤG − mod ^{0}.

Step 3. By 2.19 b) it follows that if the KSA - theorem holds for ℤG− mod ^{0} then Ḡ ≅ C_{p} with p ≤ 19 or Ḡ ≅ C_{2} × C_{2}. Note for this that if Cl(ℤG) ≠ 0 then the KSA - theorem does not hold because for a non - free projective ideal J of ℤG it follows from 2.10 b) that

J ⊕ J ≅ ℤG ⊕ I.

By 2.10 a) J, ℤG and I are indecomposable.

Step 4. If G is a p - group with Ḡ = C_{p} then G = C_{p}. If G is a 2 - group with Ḡ = C_{2} × C_{2} then G is generated by two elements. By 2.19 c) we see that G has to be a dihedral group. This completes the proof.

3. Descriptions of group rings

The aim of this section is the construction of rings which are closely related to a given integral group ring SG.

3.1. Small group rings.

3.1. Augmentation ideals. The map ε : RG ⟶ R defined by

∑_{g∈G} r_{g}g ↦ ∑_{g∈G} r_{g}

is an R - algebra homomorphism and is called the augmentation. Its kernel is called the augmentation ideal of RG and will be denoted by I(RG). In the case when R = ℤ we simply write I(G).

Lemma 3.2. (i) H is a normal subgroup of G if, and only if, RG · I(RH) is a 2 -

sided ideal. (ii) I(G)/I(G)^{2} ≅ \overline{G} (iii) ℤG · I(H)/I(G) · I(H) ≅ \overline{H} for each normal subgroup H of G. (iv) A surjective group homomorphism ϕ : G ⟶ \overline{G} induces uniquely an R - algebra

epimorphism ϕ̃: RG ⟶ R\overline{G}. If N = Kerϕ then the following sequence is exact.

0 ⟶ RG · I(RN) ⟶ RG ⟶ R\overline{G} ⟶ 0

Definition 3.3. Let N be a normal subgroup of G. Then the quotient

S_{R}(G, N) := RG/I(RG) · I(RN)

is called the small group ring of RG with respect to N. For R = ℤ we write S(G, N) instead of S_{ℤ}(G, N).

Lemma 3.4. Let A be an abelian normal subgroup of G. Let Ã, G̃, V (S(G, A)) be the images of A, G, V (ℤG) rsp. in U(S(G, A)) under the reduction map from ℤG to S(G, A)). Then there are exact sequences of groups

1 ⟶ Ã ⟶ U(S(G, A)) ⟶^{κ} U(ℤ(G/A)) ⟶ 1

and

1 ⟶ Ã ⟶ V (S(G, A)) ⟶ V (ℤ(G/A)) ⟶ 1.

Moreover Ã is isomorphic to A and κ(G̃) = G/A.

Theorem 3.5 (G. Higman). If A is an abelian finite group. Then the torsion units of V (ℤA) coincide with A.

Theorem 3.6 (Whitcomb). [64] Let G be a finite metabelian group, then ℤG ≅ ℤH ⇒ G ≅ H, i.e. the isomorphism problem has a positive answer for finite metabelian groups.

For the proof we need the following basic result.

A COURSE ON INTEGRAL GROUP RINGS 137

Lemma 3.7. Assume that ℤG and ℤH are isomorphic as rings. Then exists an isomorphism τ : ℤG ⟶ ℤH such that τ(V (ℤG)) = V (ℤH). In particular H is isomorphic to a subgroup of V (ℤG).

Proof. (of Theorem 3.6) By Lemma 3.7 we may assume that H ≤ V (ℤG) and that ℤH = ℤG. Let κ : ℤG ⟶ ℤ(G/A) be induced from the reduction map from G onto G/A, where A is an abelian normal subgroup of G such that G/A is abelian. Then κ(H) = G/A by Theorem 3.5. Hence ℤG · I(A) = ℤH · I(B) for some normal subgroup B of H. Now apply Lemma 3.4. Note that S(G, A) = S(H, B). It follows that G̃ = H̃ and therefore G ≅ H. q.e.d.

3.2. Orders. Note ℤG ⊂ ℚG ⊂ ℂG. Thus it is a natural question how these inclusions may be described.

Definition 3.8. Let R be a Dedekind domain and let K be its quotient field. Let A be a finite dimensional K - algebra. An R - order Λ in A is a subring of A with the following properties.

• R = R · 1 ⊂ Λ

• Λ is as R - module finitely generated

• Λ contains a K - basis of A.

Clearly ℤG is a ℤ - order in ℚG.

3.9. Maximal orders. Let R and A be as above. A maximal R - order in A is an R - order which is not properly contained in any larger R - order in A. The basic results on maximal orders are as follows.

a) Any R - order in A is contained in at least one maximal order R - order.

b) Maximal orders exist. An R - order Λ is called hereditary, if each left ideal of Λ is projective as Λ - module.

c) Maximal orders are hereditary.

Example 3.10. • If K is an algebraic number field then the ring R of algebraic integers of K is a maximal order in K. Moreover M_{n}(R) is a maximal order in M_{n}(K).

• RG is a maximal order in KG if, and only if, |G| is a unit in R.

Lemma 3.11. Let |G| = n. Then RG ⊂ Λ ⊂ n^{−1}RG for any R - order Λ containing RG.

3.12. Let K be an algebraic number field which is a splitting field for G and denote by R its ring of algebraic integers. Then

KG = M_{n_{1}}(K) × … × M_{n_{h}}(K).

138 WOLFGANG KIMMERLE

The projection of RG onto M_{n_{i}}(K) yields an order Λ_{i} in M_{n_{i}}(K). Clearly

RG ⊂ M_{n_{1}}(R) × … × M_{n_{h}}(R)

and by the previous lemma

|G| · M_{n_{1}}(R) × … × |G| · M_{n_{h}}(R) ⊂ RG

Example 3.13.

ℚS_{3} = ( \begin{matrix}ℚ & ℚ \\ ℚ & ℚ \end{matrix} ) × ℚ × ℚ.

Λ = ( \begin{matrix}ℤ & ℤ \\ ℤ & ℤ \end{matrix} ) × ℤ × ℤ.

ℤS_{3} consist of all tuples

( \begin{matrix}a & b \\ c & d \end{matrix} ) × (e) × (f) ∈ ( \begin{matrix}ℤ & 3ℤ \\ ℤ & ℤ \end{matrix} ) × ℤ × ℤ.

such that a ≡ e mod 3, d ≡ f mod 3 and e ≡ f mod 2. This may be easily shown with the representation given in Example 2.2 .

6Λ = ( \begin{matrix}6ℤ & 6ℤ \\ 6ℤ & 6ℤ \end{matrix} ) × 2ℤ × 2ℤ.

3.3. Pullbacks.

Definition 3.14. Let R, S_{1}, S_{2}, S_{12} be rings and let β_{1}, β_{2}, ϕ_{1}, ϕ_{2} be ring homomorphisms such that the diagram

R ⟶^{β_{1}} S_{1}
β_{2} ↓  ↓ ϕ_{1}
S_{2}  ⟶^{ϕ_{2}} S_{12}

is commutative. Then (R, β_{1}, β_{2}) is called pullback of (S_{1}, S_{2}, S_{12}, ϕ_{1}, ϕ_{2}), if for each ring T with ring homomorphisms α_{i} : T ⟶ S_{i} satisfying α_{1} · ϕ_{1} = α_{2} · ϕ_{2} there exists a unique ring homomorphism κ : T ⟶ R such that κ · β_{i} = α_{i}. One says also simply that the diagram is a pullback.

Example 3.15. a) Let N be a normal subgroup of G and assume that |N| ≠ 0 in

R. Then

RG ⟶ RG/ < N̂ >
↓  ↓
RG/N ⟶ (R/|N|R)G/N

is a pullback, where N̂ = ∑_{x∈N} x as element of RG.

b) Let (S_{1}, S_{2}, S_{12}, ϕ_{1}, ϕ_{2}) be as in Definition 3.14 then its pullback is given by R = {(x, y) ∈ S_{1} × S_{2}; ϕ_{1}(x) = ϕ_{2}(y)} together with the projections π_{i} : R ⟶ S_{i}.

A COURSE ON INTEGRAL GROUP RINGS 139

c) Put G = C_{p} then a) becomes with R = ℤ and N = G

ℤC_{p} ⟶ ℤC_{p}/ < Ĉ_{p} >= ℤ[ζ] = R
↓  ↓
ℤ ⟶  (ℤ/p · ℤ)

In computing class groups there are associated to such pullbacks so - called Mayer - Vietoris sequences which give in this case the following exact sequences of groups

1 → U(ℤC_{p}) ⟶ U(ℤ) × U(R) ⟶^{h} U(ℤ/pℤ) ⟶ Cl(ℤC_{p}) ⟶ Cl(ℤ) ⊕ Cl(R) → 0

Now U(ℤC_{p}) = C_{p} × C_{2} and Cl(ℤ) = 0. Show that h is surjective then Cl(ℤC_{p}) ≅ Cl(R) and Rims theorem 2.19 a) follows.

Further information on class groups is given e.g. in [10, Chp.6], [48], [40].

Lemma 3.16. Let M and N be normal subgroups of G. Assume that m = |M| and n = |N| are relatively prime. Then

ℤG  ⟶^{α_{1}} Γ
α_{2} ↓  ↓ β_{1}
Λ := ℤG/ < M̂, N̂ > ⟶^{β_{2}}  Λ̄

is a pull back diagram, where

Λ̄ = (ℤ/mℤ(G/M))/ < ¯ N̂ > ⊕(ℤ/nℤ(G/N))/ < ¯ M̂ >

and Γ is the image of α_{1} : ℤG/M⊕ℤG/N defined by x ↦ (x+I(M)ℤG, x+I(N)ℤG). α_{2} is the reduction map and β_{1} is induced from the obvious surjection from ℤG/M ⊕ℤG/N onto Λ̄.

3.17. Pullback approximation of group rings . Let M and N be normal subgroups of G of coprime order m, n rsp. Then there are pull back diagrams

\begin{matrix}ℤG & ⟶ & Γ \\ ↓ & & ↓ \\ Λ & ⟶ & Λ̄ \end{matrix}

Γ ⟶ ℤG/M
↓  ↓
ℤG/N ⟶ ℤG/M · N

ℤG/M ⟶ (ℤG/M)/ < N̂ >
↓  ↓
ℤG/M · N ⟶ (ℤ/nℤ)(G/M · N)

ℤG/N ⟶ (ℤG/N)/ < M̂ >
↓  ↓
ℤG/M · N ⟶ (ℤ/mℤ)(G/M · N)

3.18. Applications to automorphisms and isomorphisms.

a) Automorphisms. σ ∈ AutℤG is called normalized, if it preserves the augmenta-

tion. Aut_{n}(ℤG) denotes the group of normalized automorphisms.

Let M and N be normal subgroups of G with M ∩N = 1. Consider the following pullbacks of groups and rings.

G ⟶^{ψ_{M}}  G/M
ψ_{N} ↓  ↓ ϕ_{N}
G/N ⟶^{ϕ_{N}} G/(M · N)

and

Γ ⟶  ℤG/M
↓  ↓
ℤG/N ⟶ ℤG/(M · N)

Let σ ∈ Aut_{n}ℤG. Assume that I(M) · ℤG and I(N) · ℤG are characteristic and that normalized automorphisms of ℤG/M, ℤG/N rsp. admit a Zassenhaus decomposition. Thus σ induces normalized automorphisms σ_{Γ}, σ_{M}, σ_{n} rsp. of Γ, ℤG/M, ℤG/N rsp. and we may write σ_{M} = γ_{M} · α_{M} and σ_{N} = γ_{N} · α_{N}, where γ_{M}, γ_{N} rsp. are central and α_{M}, α_{N} are induced from group automorphisms. Thus we get along the pullback diagram of the rings the following picture

σ_{Γ}  ⟶ σ_{M} = γ_{M} · α_{M}
↓  ↓
σ_{N} = γ_{N} · α_{N} ⟶  σ̄

Moreover σ̄ = γ̄_{N} · ᾱ_{N} = γ̄_{M} · ᾱ_{M}. Thus ρ := γ̄_{M}^{−1}γ̄_{N} = ᾱ_{N}ᾱ_{M}^{−1} is a conjugacy class preserving group automorphism of G/(M · N).

A COURSE ON INTEGRAL GROUP RINGS 141

Assume now that ρ lifts to a group automorphism α of G, i.e there exist group automorphisms α_{M}, α_{N} of G/M, G/N rsp. such that the following diagram induced from the pullback diagram of groups is commutative

\begin{matrix}α & ⟶ & α_{M} \\ ↓ & & ↓ \\ α_{N} & ⟶ & ρ \end{matrix}

Then σ may be modified by α to σ̃ such that σ̃_{Γ} is central. Thus we obtained that the analogue of the Zassenhaus conjecture (ZC 2) is valid for Γ or in other words one could say that σ admits on the level of Γ a Zassenhaus decomposition. It then remains to check whether this property may be lifted from Γ to ℤG. For this the pullback

\begin{matrix}ℤG & ⟶ & Γ \\ ↓ & & ↓ \\ Λ & ⟶ & Λ̄ \end{matrix}

is of relevance. The conjecture (ZC 2) does not hold for all groups. The method described above however may also be used to construct counterexamples to (ZC 2), cf. section 5.2.

b) Isomorphisms. Assume now that σ is a normalized isomorphism between ℤG and ℤH. The normal subgroups M and N of G correspond to normal subgroups K and L of H, cf. 4.6 (iv). In the case when M and N are unique by its orders the same holds for K and L. Moreover the ideals ℤG·I(M), ℤG·I(N), ℤH ·I(K), ℤH ·I(L) are characteristic.

Suppose now that the conjecture (ZC 2) holds for ℤG/M, ℤG/N rsp. Then this holds for ℤH/K, ℤH/L and we may even identify ℤG/M with ℤH/K and ℤG/N with ℤH/L. Thus σ yields at this level normalized automorphism σ_{M}, σ_{N} as in part a). Analogously to part a) we get the conjugacy class preserving automorphism ρ. Now we may form the pullback

G(ρ) ⟶^{ψ_{M}}  G/M
ψ_{N} ↓  ↓ ϕ_{N}
G/N ⟶^{ϕ_{N} ·ρ} G/(M · N)

and an easy calculation shows that H is isomorphic to G(ρ). Moreover, if ρ lifts to an automorphism of G, then G ≅ G(ρ) and the isomorphism problem has a positive solution for G. This shows that we may regard ρ as an obstruction to the isomorphism problem. The same game may be played with more than two normal subgroups and leads to the description of G as projective limit of quotients and to a Čech - style cohomology obstruction theory, see 5.3.

4. Torsion units

4.1. The 1 - coefficient. Let x = ∑_{g∈G} r_{g}g be an element of RG. Then r_{1} is called the 1 - coefficient of x and is denoted by tr(x).

Theorem 4.1 (Saksonov). [54] Let S be a G - adapted integral domain of characteristic zero and let x be a torsion element of V (SG). Then tr(x) = 0 unless x = 1.

Corollary 4.2. Let S be a G - adapted integral domain of characteristic zero with quotient field K and let H ≤ V (SG).

• H consists of R - linearly independent elements.

• |H| = |G| ⇒ SG = SH, i.e. H is a group basis of SG.

• |H| divides |G| and KG is as KH - module free.

• RH is a pure R - sublattice of RG.

Corollary 4.3. Assume that π ⊃ π(G). Then U(ℤ_{π}G) ≅ U(ℤ_{π}H) ⇒ ℤ_{π}G ≅ ℤ_{π}H.

The antipode ∗ : RG ⟶ RG of a group ring RG is defined as the R - linear map induced from g ↦ g^{−1}, i.e.

x = ∑_{g∈G} r_{g}g ↦ x^{∗} = ∑_{g∈G} r_{g}g^{−1}.

Corollary 4.4 (Banaschweski). ℤG and its antipode determine G up to isomorphism.

Proof. Let τ : ℤG ⟶ ℤH be an isomorphism such that τ ∘ ∗ = ∗ ∘ τ. For convenience we write for the antipode of ℤG and for that one of ℤH just *. As in lemma 3.7 one sees that we may assume that τ preserves the augmentation. Now

τ(G)^{∗} = τ(G^{∗}) = τ(G)

and τ(G) is fixed by the antipode of ℤH. Let x be a normalized unit of ℤH. Then tr(xx^{∗}) > 0 and tr(xx∗) = 1 if, and only if x ∈ H. Hence H is the only group basis of ℤH fixed by its antipode. Thus it follows that τ(G) = H and G and H have to be isomorphic. q.e.d.

4.2. Characters.

Theorem 4.5 (Saksonov). Let S be a G - adapted integral domain and let H be a group basis of SG.. Then exists a bijection σ : G ⟶ H such that

• Ĉ_{g} = { \atop C_{σ(g)}}

• { \atop C_{σ(g^{n})}} = { \atop C_{(σ(g))^{n}}}.

A COURSE ON INTEGRAL GROUP RINGS 143

Remark 4.6. (i) A bijection σ with the properties of Theorem 4.5 is called a class sum correspondence between G and H or more generally between group bases of SG.

(ii) If between group bases H_{1} and H_{2} a class sum correspondence exists which is additionally a homomorphism then H_{1} and H_{2} are conjugate in KG, where K denotes the quotient field of S. Thus the Zassenhaus conjecture (ZC 2) may be phrased also in the form that there always exists a class sum correspondence which is a homomorphism or in other words that two group bases are linked by an isomorphism over the class sums.

(iii) Theorem 3.5 is a special case of Theorem 4.5

(iv) If N is a normal subgroup of G and σ is a class sum correspondence between G and H then σ(N) is a normal subgroup of H. This yields a normal subgroup correspondence which is a lattice isomorphism between the normal subgroup lattices of G and H.

Corollary 4.7 (Glauberman). Let S be a G - adapted integral domain. Then SG ≅ SH implies that G and H have the same spectral table ^{4}.

σ ∈ Aut_{n}(ℤG) induces via the class sum correspondence on the spectral table ST (G) of G a table automorphism. Denote by Aut(ST (G) the group of table automorphisms and by Aut_{ℤ}(ST (G)), Aut_{G}(ST (G)) rsp. the subgroups induced by Aut_{n}(ℤG), AutG rsp.

Proposition 4.8. a) (AUT) holds if and only if Aut_{ℤ}(ST (G)) = Aut_{G}(ST (G)). b) Aut(ST (G)) = Aut_{G}(ST (G) ⇒ (AUT).

Example 4.9. Consider the character table of A_{5}.

( \begin{matrix}1 & 1 & 1 & 1 & 1 \\ 3 & −1 & 0 & a & ā \\ 3 & −1 & 0 & ā & a \\ 4 & 0 & 1 & −1 & −1 \\ 5 & 1 & −1 & 0 & 0 \end{matrix} )

There is obviously just one non-trivial table automorphism interchanging the two characters of degree 3 and the classes of elements of order 5. This automorphism is induced

4The spectral table of a finite group is its character table including the headline, i.e. in particular including the power map on the classes

144 WOLFGANG KIMMERLE

by an outer automorphism of A_{5} , e.g. by conjugation with a transposition of S_{5}. Clearly the character table automorphisms coincide in this example with the spectral table automorphisms. By Proposition 4.8 b) it follows that (AUT) holds for ℤA_{5}.

4.10. Modular characters. Usually the spectral table does not give sufficient information in order to determine Aut_{ℤ}(ST (G)). More information is given if one considers additionally the action of Aut_{n}(ℤG) on the modular representations of G.

Note, if (K,R,k) is a p - modular system for G (cf. I, 2.7), sufficiently large, then σ ∈ Aut_{n}(ℤG) extends uniquely to an R - algebra automorphism of RG and induces then automorphisms of KG, kG rsp. In particular one obtains naturally an action on the p - modular representations of G. If the Sylow p - subgroups are cyclic, then the principal p - blocks are Brauer tree algebras. In [6] it is shown that each autoequivalence of such a block, which fixes the trivial simple module, fixes the isomorphism class of each simple module. Indeed even the isomorphism class of each indecomposable module will be fixed in this situation [3]. Since a normalized automorphism of ℤG fixes ℤ it induces such an autoquivalence of the principal block and this information is very useful in order to calculate Aut_{ℤ}(ST (G)). Similar considerations can be made with other cyclic blocks provided they are fixed.

In the case when the blocks do not have cyclic defect then the action on the decomposition matrix (cf. I 4.11) still gives additional information. These modular methods may be applied to simple groups of Lie type of small rank and lead often to a proof of (ZC 2), cf. [4].

Closely related to these modular methods is the so - called defect group problem, i.e. the question whether independent of G the defect group of a block of ℤ_{p}G is unique up to conjugation and suitable normalization [57]. For progress on this question see [33].

Example 4.11. The Mathieu group M_{23}. Note |M_{23}| = 2^{7} · 3^{2} · 5 · 7 · 11 · 23.

The ordinary character table can be found in [9] or may be considered with GAP [55]. The parts we need are shown below. M_{23} has 17 conjugacy classes. At the conjugacy classes (2a,3a,4a,5a,6a,8a) which are omitted the displayed characters of the same degree have the same value. The remaining irreducible characters have degree 1, 22, 230, 231, 1035, 2024. The character values of the missing character of degree 231 are all rational. A, /A, B … denote in the usual way irrationalities.

( \begin{matrix}& 1a & 7a & 7b & 11a & 11b & 14a & 14b & 15a & 15b & 23a & 23b \\ X.3 & 45 & A & /A & 1 & 1 & −A & −/A & . & . & −1 & −1 \\ X.4 & 45 & /A & A & 1 & 1 & −/A & −A & . & . & −1 & −1 \\ X.7 & 231 & . & . & . & . & . & . & C & /C & 1 & 1 \\ X.8 & 231 & . & . & . & . & . & . & /C & C & 1 & 1 \\ X.10 & 770 & . & . & . & . & . & . & . & . & D & /D \\ X.11 & 770 & . & . & . & . & . & . & . & . & /D & D \\ X.12 & 896 & . & . & B & /B & . & . & 1 & 1 & −1 & −1 \\ X.13 & 896 & . & . & /B & B & . & . & 1 & 1 & −1 & −1 \\ X.14 & 990 & A & /A & . & . & A & /A & . & . & 1 & 1 \\ X.15 & 990 & /A & A & . & . & /A & A & . & . & 1 & 1 \end{matrix} )

The character table automorphisms (given as permutations on the irreducible characters) of M_{23} are given by

< (χ_{3}, χ_{4})(χ_{14}, χ_{15}), (χ_{7}, χ_{8}), (χ_{10}, χ_{11}), (χ_{12}, χ_{13}) >

where the ordering of the characters is according to [9]. The examination of principal blocks with cyclic defect shows that a normalized automorphism of ℤM_{23} fixes all these ordinary characters except possibly the characters χ_{10} and χ_{11}. Note that an irreducible character χ belongs to the principal p - block if and only if

\frac{χ(g)}{χ(1)} · |Cl(g)| ≡ |Cl(g)| mod p

for each p - regular element g ∈ G. |Cl(g)|denotes the length of the conjugacy class of g.

The decomposition matrix of the non - cyclic principal 3-block gives additional information ([21, Theorem 7.4.]).

The picture of the corresponding component of the Brauer graph of this block is shown below:

[figure: 10; 12; 2; 1; 17; 5; 9; 13; 11]

If there existed a normalized algebra automorphism of the integral group ring permuting the characters χ_{10} and χ_{11}, then it would also permute the characters χ_{12} and χ_{13}. But we already know that these characters are fixed. Thus every normalized algebra automorphism of the integral group ring fixes every ordinary irreducible character. Thus we have established that (AUT) is valid for ℤM_{23}.

4.3. Normalizers.

Problem 4.12. Let V = V (ℤG), U ≤ G, x ∈ N_{V} (U).

a) Is there g ∈ N_{G}(U) such that the action xg^{−1} centralizes U ?

b) In the case when U = G. Is N_{V} (G) = G · C_{V} (G) ? Both questions will also be considered when ℤ is replaced by a G - adapted domain R.

Lemma 4.13 (Coleman). Let G be a finite group, let P a p-subgroup of G and let S be an integral domain, in which the prime p is not a unit. Let V = V (SG). Then

N_{V} (P) = N_{G}(P) · C_{V} (P).

Definition 4.14. Denote by Aut_{R}(G) the group of all automorphisms of G which become inner on RG.

Remark 4.15. • Aut_{ℂ}(G) coincides precisely with the conjugacy class preserving

automorphisms of G.

• The question whether Aut_{ℤ}(G) = InnG is equivalent to the normalizer problem 4.12 b).

A COURSE ON INTEGRAL GROUP RINGS 147

• Note that by Colemans lemma σ ∈ Aut_{ℤ}(G) has the property that for each Sylow subgroup S of G exists an inner automorphism γ_{S} such that γ_{S} · σ is the identity on S.

4.16. Results on the normalizer problem.

a) (Jackowski - Marciniak) [20] If G has a normal Sylow 2 - subgroup then Aut_{ℤ}(G) = InnG.

b) (Krempa) [20] Aut_{ℤ}(G)/InnG is an elementary abelian 2 - group.

c) Part a) implies in particular that for groups of odd order the normalizer problem has a positive solution. This holds more general. If G has no composition factor of order 2, then Aut_{ℤ}(G) = InnG. [27]

Remark 4.17. It was shown in [50] that there are groups such that outer group automorphisms become inner on ℤ_{π}G. Recently M. Hertweck constructed a global counterexample [17], i.e. one for ℤG counterexamples. It is an important ingredient for his counterexample to the isomorphism problem, see subsection 4.5.

The results of [50] combined with recent results of Mazur [35] yield in a different way a counterexample to the isomorphism problem for infinite groups [51]. The coefficient ring in this counterexample however is the ring of algebraic integers in a suitable algebraic number field and not ℤ.

4.4. Application to Hecke algebras. Let H be the generic Iwahori - Hecke algebra (cf. IV, 5C) over the ring A = ℤ[v, v^{−1}] of Laurent polynomials in one indeterminate associated to the Coxeter group W. This algebra may be viewed as a deformation of the integral group ring of W. The A - algebra H is a free A-module with basis {T_{w} | w ∈ W }, and the multiplication is given by the rules

\begin{aligned}T_{w}T_{w′} & = & T_{ww′} & if l(ww′) = l(w) + l(w′), \\ T_{s}^{2} & = & uT_{1} + (u − 1)T_{s} & for s ∈ S \end{aligned}

where u := v^{2} and l: W → ℕ_{0} is the usual length function on W. (For the construction of this algebra see [10, section68A], for example.)

The 1-dimensional representation of H given by ind: T_{s} ↦ u (s ∈ S) is called the index representation of H. An arbitrary A-algebra automorphism σ: H → H will be called normalized if ind ∘ σ = ind.

The specialization v = 1 yields ℤW as an image of H and the knowledge of the automorphisms of ℤW gives a crucial information about that ones of H.

Theorem 4.18. [5] Let W be a finite Coxeter group then (ZC 2) holds for ℤW.

The proof of Theorem 4.18 is obtained with Proposition 4.8 by showing that Aut(ST (W)) = Aut_{W} (ST (W)).

148 WOLFGANG KIMMERLE

Theorem 4.19. Let H be a generic Hecke algebra with Coxeter diagram Δ. If σ is a normalized automorphism of H then σ is the composition of a graph automorphism of Δ and a central automorphism.

This result shows that Hecke algebras are even more rigid than integral group rings of the Coxeter groups. In order to explain this we look at the example of the Hecke algebra of type A_{5}. The corresponding Coxeter group is S_{6}. All graph automorphisms of Hecke algebras of type A_{n} are central and thus by Theorem 4.19 all normalized automorphisms of H are central. On the other hand the symmetric group S_{6} has a well known non-trivial outer automorphism. We remark that in order to show that such an automorphism does not exist on the level of the Hecke algebra one uses in addition to the specialization to ℤW other specializations of H and looks at blocks of these specializations which are Brauer trees. For the details we refer to [5].

4.5. Construction of units. 1997 M. Hertweck [17] solved the isomorphism problem by constructing two non - isomorphic finite groups with isomorphic integral group rings. In this subsection some of the basic techniques for this will be discussed.

4.20. M. Hertwecks idea for the counterexamples is as follows.

• Construct a group G with a non - inner automorphism τ such that τ becomes inner on ℤG, i.e. there is an element t ∈ ℤG with gτ = g^{t} for each g ∈ G.

• Let X be the semi - direct product of G ×C, where C is a cyclic group generated by c. In ℤX the equation t^{c} = t^{−1}. is satisfied.

• Put Y =< G, tc >≤ V (ℤX).

M. Hertweck showed that such a construction can be done with groups X, Y of order 2^{21} · 97^{28} and X is not isomorphic to Y.

Remark 4.21. Let t and c be as above. Then tctc = c^{2} has finite order. If c is a 2 - element then it is clear that tc has the same order as c. Clearly tc normalizes G and it follows that Y has the same order as X. Now by Corollary 4.2 it follows automatically that ℤX = ℤY. This means that the isomorphism of the group rings is garanteed by the construction. Thus the main task is now to carry out the construction such that X and Y are non - isomorphic. This is mainly a group - theoretical question. For this the obstruction theory explained in 3.18 is helpful and in particular the pullback approximation for group rings , cf. subsection 3.3, is used.

4.22. Next we explain by an exemplary computation how pullbacks may be used to construct units. The example shows also why the prime 97 is involved in the order of Hertwecks counterexamples. Let C_{8} =< a > . In ℤC_{8} consider

ε = \frac{1}{2}(1 + a^{4}), η = \frac{1}{2}(1 − a^{4}).

Then v = ε + η(3 − 2(a + a^{−1})) is a unit. More general

u = ε + η(x_{n} + y_{n}(a + a^{−1}))

is a unit provided x_{n}^{2} − 2y_{n}^{2} = 1.

The aim is now to construct a group G with the following properties. G has the following structure. G = (L × Q) × C, where L and Q are groups of coprime prime power order. Let q = |Q|, l = |L|. According to Example 3.15 we get the following pullbacks of rings. For simplicity we denote the image of Q, C rsp. in G/L again by Q, C rsp.

ℤG/L ⟶ (ℤG/L)/ < Q̂ >
κ_{L} ↓  ↓
ℤG/(L · Q) ⟶ (ℤ/qℤ)G/(L · Q)

Changing the role of L and Q we get analogously

ℤG/Q ⟶ (ℤG/Q)/ < L̂ >
κ_{Q} ↓  ↓
ℤG/(L · Q) ⟶ (ℤ/lℤ)G/(L · Q)

Now ℤG/L should have a unit w_{L} such that κ_{L}(w_{L}) = a^{4} · u^{2} and κ_{Q}(w_{Q}) = a^{4} · u^{2}. Note, if w_{L}.w_{Q} are constructed, we get via the pullback

Γ ⟶ ℤG/Q
↓  ↓
ℤG/L ⟶ ℤG/(L · Q)

a unit w in Γ which maps onto a^{4} · u^{2} in ℤG/(L · Q).

A nice calculation shows that

u^{2} ≡ a^{4} mod p^{m}, u^{2} ≡ 1 mod 2^{k}

if p^{m} divides x_{n} and 2^{k} divides y_{n}. Note

1 = x^{2} − y^{2} = (x + y\sqrt{2})(x − y\sqrt{2})

and

1 = 1^{m} = (x + y\sqrt{2})^{m}(x − y\sqrt{2})^{m}.

Use 1 = (3 + 2\sqrt{2})(3 − 2\sqrt{2}). as start for the calculation. Then the smallest m such that (3 + 2\sqrt{2})^{m} = x̃ + ỹ\sqrt{2} with 8|ỹ is m = 12. Then x̃ = 97 · x.

150 WOLFGANG KIMMERLE

Assume now that Q has order 8. Then we may choose |L| as a power of 97 and we get finally the units w_{L} and w_{Q} along the pullback diagrams for ℤG/L, ℤG/Q rsp.

w_{L} ⟶ a^{4} · u^{2}
↓  ↓
a^{4} ⟶ a^{4}

and

w_{Q} ⟶ a^{4} · u^{2}
↓  ↓
1 ⟶ 1

5. Survey

5.1. Results about specific classes of finite groups. Let SG be an integral group ring of a finite group G.

• (CP) is valid for SG, if G is abelian. [19] [54]

• (IP) is valid for SG, if G is metabelian [64]

• (ZC 2) is valid for SG, if G is nilpotent [45] [61] (CP) holds for ℤ_{p}G, if G is a p - group. [61]

• (ZC 3) is valid for SG, if G is nilpotent. [62]

• (IP) is valid for simple groups and their automorphism groups [28]

• (ZC 1) is valid for metacyclic groups. [34]

• (IP) is valid for nilpotent by abelian groups [56] [49]

• (ZC)_{π(G)} is valid for abelian by nilpotent groups. Note that this class of groups contains the supersoluble groups. [23],[25]

• (ZC 2) is valid for finite Coxeter groups [5]

• (IP) holds for finite complex reflection groups. [15]

• (ZC 2) is valid for all minimal simple groups, for simple groups of Lie type of small rank and most of the sporadic groups. [2] [4], [6]

5.2. Counterexamples.

• It is an easy exercise to see that (CP) fails for ℤS_{3}, where S_{3} denotes the symmetric group of degree 3. There exists an involution t which is not contained in a group basis.

• (ZC 2) does not hold for ℤG for all metabelian groups [47] [32]. A semi-local counterexample for A - groups has been constructed in [16]. For other semi - local counterexamples for groups of small order see [1].

• (IP ) does not hold in general for finite nilpotent by nilpotent groups. [17]

A COURSE ON INTEGRAL GROUP RINGS 151

5.3. Structural results.

Theorem 5.1. [28] Assume that the integral group rings SG and SH are isomorphic. Then G and H have the same chief series.

Theorem 5.2. [46] [56] Let G be a finite group and assume that the generalized Fitting subgroup F^{∗}(G) is a p - group. Then (ZC 2) is valid for the integral group ring SG.

The proof of Theorem 5.2 is not completely published. Also the original proof, cf. the sketch in [44], has a gap in the case when p = 2, which however could be filled out in [18]. Theorem 5.2 leads to an obstruction theory in Čech cohomology style, similar to the covering of topological spaces by open sets. It is in particular useful for soluble groups because the assumption of Theorem 5.2 is for soluble groups satisfied if O_{p}(G) = 1 for some prime p.. The cocycles ρ) of this cohomology are tuples consisting of conjugacy class preserving automorphisms of certain proper quotients of G. They permit to define a group G(ρ). ρ will be a coboundary if it is induced from a group automorphism of G.In 3.18 we considered the special case when G is described as a pullback.

Theorem 5.3. [29] Let G be a finite soluble group and assume that ℤG is isomorphic to ℤH. Then there exists a cocycle ρ such that H ≅ G(ρ). Moreover H is isomorphic to G if and only if ρ is a coboundary.

With respect to Sylow subgroups the following is known.

Theorem 5.4. [30] Let G be a finite soluble group. Then (Syl)_{1} is valid, i.e. if H is a group basis of ℤG then each Sylow subgroup of H is conjugate within ℚG to a Sylow subgroup of G.

Theorem 5.5. [12] (Syl)_{2} holds for finite nilpotent - by - nilpotent groups. Frobenius groups.

(Syl)_{2} is also known for Frobenius groups which do not have S_{5} as image [13].

Theorem 5.6. [31] Assume that ℤG ≅ ℤH. Then abelian Sylow subgroups of G and H are isomorphic.

Note that the last result already holds at the level of character tables and gives a positive answer to one of the famous problems of R. Brauer [8]. All these problems circle around the isomorphism problem and deal with the question which properties of a finite group are reflected by its representations.

For further reading on group rings, especially integral group rings, the following books and survey articles are mentioned:

[38], [48] , [59], [7] [53], [26], [42], [44], [57], [41], [36], [63].

6. List of some notations

Let G be a finite group

π(G) set of primes dividing |G|
|G|  order of G
[G, G] commutator subgroup
\overline{G}  G/[G, G]
C_{n}  cyclic group of order n
S_{n}  symmetric group of degree n
A_{n}  alternating group of degree n
D_{n}  dihedral group of order 2n

Let R be a ring,let G be a group. Let H be a subgroup of G and let M be subset of G.

U(R)  unit group of R
I(RG)  augmentation ideal
I(G)  augmentation ideal of ℤG
RG/H  permutation module relative to H
RG · I(RH) left ideal generated by I(RH)
V (RG)  units of augmentation 1
M̂  ∑_{g∈M} g ∈ RG, the trace of M.
ℤ_{p} \{\frac{a}{b} ∈ ℚ; (a, b) = 1, p does not divide b\}
ℤ_{π} ⋂_{p∈π} ℤ_{p}
F_{q} field of q elements

References

[1] , P. F. Blanchard, Exceptional group ring automorphisms for some metabelian groups, Comm.Alg. 25 (1997), Nr.9, 2735 – 2742.

[2] F. M. Bleher, Zassenhaus Vermutung und einfache Gruppen, Diplomarbeit Universität Stuttgart 1993.

[3] , Automorphismen von Gruppenringen und Blocktheorie, Dissertation, Stuttgart 1995.

[4] , Tensor products and a Conjecture of Zassenhaus, Arch. Math., Vol.64, 289–298 (1995).

A COURSE ON INTEGRAL GROUP RINGS 153

[5] F. M. Bleher, M. Geck and W. Kimmerle, Automorphisms of generic Iwahori - Hecke algebras and integral group rings of finite Coxeter groups, J. of Algebra 197 (1997), 615–655.

[6] F. M. Bleher, G. Hiss and W. Kimmerle, Autoequivalences of Blocks and a Conjecture of Zassenhaus, J. of Pure and Appl. Algebra 103 (1995) 23-43.

[7] A. A. Bovdi, Group Rings (2nd edition), Uchebno-metodicheskij kabinet Visheho Obrazovanija, Kiev, 1988.

[8] R. Brauer, Representations of finite groups, Lecture on Modern Mathematics, Wiley, New York 1963, reprinted in: R.Brauer, Collected papers, Vol.II, 183 – 225 , MIT Press, Cambridge MA 1980.

[9] J. H. Conway,R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups, Oxford University Press, London/New York, 1985.

[10] CR2 C. Curtis and I. Reiner, Methods of Representation Theory Vol. I and II, Wiley Interscience, New York 1981 and 1987

[11] E. Dade, Deux groupes finis distincts ayant la meme algebre de groupe sur tout corps, Math.Z. 119 (1971), 345 – 348.

[12] M. A. Dukuchaev and S. O . Juriaans, Finite subgroups in integral group rings, Can.J.Math. 48 (1996), Nr.6, 1170 – 1179.

[13] M. A. Dukuchaev, S. O. Juriaans and C. P. Milies, Integral group rings of Frobenius groups and the conjectures of H. J. Zassenhaus, Comm.Alg. 25 (1997), Nr.7, 2311 – 2325.

[14] S. Endo, T. Miyata, K. Sekiguchi, Picard groups and automorphism groups of integral group rings of metacyclic groups, J. of Algebra 77 (1982) 286 - 310.

[15] A.Greiner, Ganzzahlige Gruppenringe komplexer Spiegelungsgruppen, Diplomarbeit, Stuttgart 1997.

[16] M.Hertweck, Über eine Vermutung von H. Zassenhaus für Gruppenringe von endlichen Gruppen mit abelschen Sylowgruppen, Preprint 1993.

[17] , Two non-isomorphic finite groups with isomorphic integral group rings, to appear Comptes Rendus, Paris (extended version (in german): M.Hertweck, Eine Lösung des Isomorphieproblems für ganzzahlige Gruppenringe von endlichen Gruppen; Dissertation 1998, Universität Stuttgart.)

[18] M. Hertweck and W. Kimmerle, On the F^{∗} - Theorem, appears in the Proceedings of Groups 1997 Bath - St.Andrews.

[19] G. Higman, The units of group rings, Proc. London Math. Soc. (2),46 1940, 231 - 248.

[20] S. Jackowski and Z. Marciniak, Group automorphisms inducing the identity map on cohomology, J. Pure Appl. Algebra 44 (1987), 241 –250.

[21] G. James, The modular characters of the Mathieu groups. J. Algebra 27 (1973) 57–111.

[22] A. Jones, Groups with a finite number of integral representations, Michigan Math. J. 10 (1963), 257 – 261.

[23] W. Kimmerle, Beiträge zur ganzzahligen Darstellungstheorie endlicher Gruppen, Bayreuther Mathematische Schriften, 36 (1991) 1–139.

[24] , Variations of the Zassenhaus conjecture, in: [48], 91 -103.

[25] , Class Sums of p - elements, in: [48], 117 – 124.

[26] , On automorphisms of ℤG and the Zassenhaus conjectures, Can.Math.Soc. Conf.Proc. 18 (1996), 383 – 397.

[27] , On the normalizer problem of integral group rings, preprint.

154 WOLFGANG KIMMERLE

[28] W. Kimmerle, R. Lyons, R. Sandling and D. Teague, Composition factors from the group ring and Artin’s theorem on orders of simple groups, Proc. London Math. Soc. (3) 60 (1990) 89–122.

[29] W. Kimmerle and K. W. Roggenkamp, Projective limits of group rings, J. Pure Appl. Algebra 88 (1993) 119–142.

[30] , A Sylowlike theorem for integral group rings of finite solvable groups, Arch.Math. Vol.60, (1993) 1 - 6.

[31] W. Kimmerle and R. Sandling , Group Theoretic and Group Ring Theoretic Determination of Certain Sylow and Hall Subgroups and the Resolution of a Question of R. Brauer, J.of Algebra 171, (1995) 329 -346.

[32] L. Klingler, Construction of a counterexample to a conjecture of Zassenhaus, Comm. Alg., 19 (8), (1991), 2303–2330.

[33] M. Linckelmann, The isomorphism problem for cyclic blocks and their source algebras , Invent. Math. 125 (1996), Nr.2, 265 – 283.

[34] Z. Marciniak, J. Ritter, S. K. Sehgal and A. Weiss, Torsion units in integral group rings of some metabelian groups, II, J.Number Theory 25 (1987), 340 – 352.

[35] M. Mazur, On the isomorphism problem for infinite group rings, Expositiones Mathematicae, Spektrum Akademischer Verlag (Heidelberg) 13 (1995), 433 – 445.

[36] I. B. S. Passi, Group rings and their augmentation ideals, Lect. Notes in Math. 715, Springer Berlin 1979.

[37] D. S. Passman, Isomorphic Groups and Group Rings, Pacific Journal of Mathematics 15 (1965), 561-583.

[38] , The algebraic structure of group rings, Interscience New York, 1977

[39] G. L. Peterson, Automorphisms of the integral group ring of S_{n}, Proceedings AMS, Vol.59, Nr.1, (1976) 14–18.

[40] I. Reiner and K. W. Roggenkamp, Integral representations, Lect. Notes in Math. 744, Springer Berlin 1979.

[41] J. Ritter, Large subgroups in the unit group of group rings, Bayr. Math.Schr. 33 (1990), 153 – 171.

[42] J. Ritter and S. K. Sehgal, Generators of subgroups of U(ℤG), Contemporary Mathematics (93), 331 – 347.

[43] K. W. Roggenkamp, Observations to a conjecture of H. Zassenhaus. Groups St. Andrews 1989, Vol. 2, London Math. Soc. Lecture Note Series 160 (1991), 427–444.

[44] , The Isomorphism Problem for Integral Group Rings of Finite Groups, Proceedings of the Int. Congres of Mathematicians, Kyoto 1990, Springer 1991, 369-380.

[45] K. W. Roggenkamp and L. L. Scott, Isomorphisms of p-adic group rings, Annals of Mathematics 126 (1987) 593–647.

[46] , A strong answer to the isomorphism problem for finite p-solvable groups with a normal p-subgroup containing its centralizer, Manuscript 1987.

[47] , On a conjecture of Zassenhaus, Manuscript 1988.

[48] K. W. Roggenkamp and M. Taylor, Group rings and class groups, DMV-Seminar Band 18, Birkh”auser, Basel Boston Berlin 1992.

[49] K. W. Roggenkamp and A. Zimmermann, On the Isomorphism Problem for Integral Group Rings of Finite Groups, Arch. Math. Vol.59 (1992), 534 - 544.

A COURSE ON INTEGRAL GROUP RINGS 155

[50] , Outer group automorphisms may become inner in the integral group ring, J.of Pure and Appl.Algebra 103 (1995), 91 – 99.

[51] , A counterexample for the isomorphism problem of polycyclic groups, J. of Pure and Appl. Algebra 103 (1995), 101 –103.

[52] R. Sandling, Graham Higmans thesis Units in group rings, in: Integral representations and Applications. Lect.Notes in Math., 882, Springer 1981, 93 - 116.

[53] , The isomorphism problem for group rings; a survey, in: Orders and their applications. Lect.Notes in Math. 1142, Springer 1985, 256 - 288.

[54] A. I. Saksonov, On the group ring of finite groups I, Publ.Math.Debrecen 18 (1971), 187 - 209.

[55] M. Schönert et al., GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, first ed., 1992.

[56] L. L. Scott, Recent progress on the isomorphism problem, Proc. Symposia in Pure Math. Vol.47 (1987) 259–274.

[57] , Defect groups and the isomorphism problem, Représentations linéaires des groupes finis, Proc. Colloq. Luminy, France 1988, Astèrisque 181-182 (1990), 257–262.

[58] S. K. Sehgal, Torsion Units in Group Rings; Proceedings Nato Institute on Methods in Ring Theory, Antwerp, D.Reidel, Dordrecht, 1983, 497-504.

[59] , Units in integral group rings, Longman Scientific and Technical, Essex, 1993.

[60] , R. G. Swan, Induced representations and projective modules, Ann. of Math. 71 (1960), 552 – 578.

[61] A. Weiss, p-adic rigidity of p-torsion, Annals of Mathematics, 127 (1987) 317–332.

[62] , Torsion units in integral group rings, J. reine angew. Math. 415 (1991) 175–187.

[63] A.-Weiss, Rigidity of π - adic p - torsion, in [59], 309 - 329, 1993.

[64] A. Whitcomb, The group ring problem, Thesis University of Chicago, 1968.

[65] H. Zassenhaus, Vortragsauszug Brauer Invariance II, Tagungsbericht 24/1988 Mathematisches Forschungsinstitut Oberwolfach.

Math. Inst. B, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

    E-mail address: kimmerle@mathematik.uni-stuttgart.de