On the first Zassenhaus Conjecture

Transcription

1 Leo Margolis Joint work with Mauricio Caicedo and Ángel del Río University of Murcia, University of Stuttgart Darstellungstheorietage, Magdeburg, November

2 Talk Structure 1. The Zassenhaus Conjectures, some results and variations. 2. The cyclic-by-abelian case and the proof strategy. 3. Some methods used in the proof.

3 Notations and Setting G a finite group RG group ring of G over the ring R U(RG) unit group of RG ε : RG R augmentation map, ε( r g g) = g G r g g G V(RG) the units of augmentation 1 aka normalized units If R = Z, then U(ZG) = ±V(ZG), so suffices to study V(ZG)

4 Notations and Setting G a finite group RG group ring of G over the ring R U(RG) unit group of RG ε : RG R augmentation map, ε( r g g) = g G r g g G V(RG) the units of augmentation 1 aka normalized units If R = Z, then U(ZG) = ±V(ZG), so suffices to study V(ZG)

5 Notations and Setting G a finite group RG group ring of G over the ring R U(RG) unit group of RG ε : RG R augmentation map, ε( r g g) = g G r g g G V(RG) the units of augmentation 1 aka normalized units If R = Z, then U(ZG) = ±V(ZG), so suffices to study V(ZG)

6 Some basic results on torsion units General results on torsion units ( 40s- 60s): u = z g g V(ZG) of finite order, g G then z 1 0 implies u = 1 (Berman-Higman). A a finite abelian group. Then U(ZA) = ±A F, where F is torsion-free abelian. H V(ZG), H finite. Then H divides G. exp(v(zg)) = exp(g).

7 Some basic results on torsion units General results on torsion units ( 40s- 60s): u = z g g V(ZG) of finite order, g G then z 1 0 implies u = 1 (Berman-Higman). A a finite abelian group. Then U(ZA) = ±A F, where F is torsion-free abelian. H V(ZG), H finite. Then H divides G. exp(v(zg)) = exp(g).

8 Some basic results on torsion units General results on torsion units ( 40s- 60s): u = z g g V(ZG) of finite order, g G then z 1 0 implies u = 1 (Berman-Higman). A a finite abelian group. Then U(ZA) = ±A F, where F is torsion-free abelian. H V(ZG), H finite. Then H divides G. exp(v(zg)) = exp(g).

9 Some basic results on torsion units General results on torsion units ( 40s- 60s): u = z g g V(ZG) of finite order, g G then z 1 0 implies u = 1 (Berman-Higman). A a finite abelian group. Then U(ZA) = ±A F, where F is torsion-free abelian. H V(ZG), H finite. Then H divides G. exp(v(zg)) = exp(g).

10 The Zassenhaus Conjectures Conjectures (H.J. Zassenhaus, in the 60s) (ZC1): For u V(ZG) of finite order there exists x U(QG) s.t. x 1 ux G. (ZC2): For H V(ZG) with H = G there exists x U(QG) s.t. x 1 Hx = G. (ZC3): For H V(ZG) with H finite there exists x U(QG) s.t. x 1 Hx G. (Aut): ϕ Aut(ZG), then there exist α U(QG) and β Aut(G), s.t. ϕ = c α β, where c α denotes the conjugation with α. Set (IP): ZG = ZH G = H. Then: (ZC3) = (ZC2) (IP), (Aut) (ZC1) (IP) and (Aut)

11 The Zassenhaus Conjectures Conjectures (H.J. Zassenhaus, in the 60s) (ZC1): For u V(ZG) of finite order there exists x U(QG) s.t. x 1 ux G. (ZC2): For H V(ZG) with H = G there exists x U(QG) s.t. x 1 Hx = G. (ZC3): For H V(ZG) with H finite there exists x U(QG) s.t. x 1 Hx G. (Aut): ϕ Aut(ZG), then there exist α U(QG) and β Aut(G), s.t. ϕ = c α β, where c α denotes the conjugation with α. Set (IP): ZG = ZH G = H. Then: (ZC3) = (ZC2) (IP), (Aut) (ZC1) (IP) and (Aut)

12 The Zassenhaus Conjectures Conjectures (H.J. Zassenhaus, in the 60s) (ZC1): For u V(ZG) of finite order there exists x U(QG) s.t. x 1 ux G. (ZC2): For H V(ZG) with H = G there exists x U(QG) s.t. x 1 Hx = G. (ZC3): For H V(ZG) with H finite there exists x U(QG) s.t. x 1 Hx G. (Aut): ϕ Aut(ZG), then there exist α U(QG) and β Aut(G), s.t. ϕ = c α β, where c α denotes the conjugation with α. Set (IP): ZG = ZH G = H. Then: (ZC3) = (ZC2) (IP), (Aut) (ZC1) (IP) and (Aut)

13 Some results Positive results: (ZC3) for nilpotent groups (A. Weiss 91) (Aut) for S n, H S n with H abelian/p-group/s k, for direct products of satisfying groups (ZC2) for some simple groups (PSL(2, p), most sporadic simple groups), but unknown for A n Counterexamples to (ZC2) were first found by Roggenkamp and Scott ( 87) with G = Later Hertweck and Blanchard ( 99) found one of order 96. (ZC1) remains open in general and is for this reason today called The Zassenhaus Conjecture.

14 Some results Positive results: (ZC3) for nilpotent groups (A. Weiss 91) (Aut) for S n, H S n with H abelian/p-group/s k, for direct products of satisfying groups (ZC2) for some simple groups (PSL(2, p), most sporadic simple groups), but unknown for A n Counterexamples to (ZC2) were first found by Roggenkamp and Scott ( 87) with G = Later Hertweck and Blanchard ( 99) found one of order 96. (ZC1) remains open in general and is for this reason today called The Zassenhaus Conjecture.

15 Some results Positive results: (ZC3) for nilpotent groups (A. Weiss 91) (Aut) for S n, H S n with H abelian/p-group/s k, for direct products of satisfying groups (ZC2) for some simple groups (PSL(2, p), most sporadic simple groups), but unknown for A n Counterexamples to (ZC2) were first found by Roggenkamp and Scott ( 87) with G = Later Hertweck and Blanchard ( 99) found one of order 96. (ZC1) remains open in general and is for this reason today called The Zassenhaus Conjecture.

16 Some results Positive results: (ZC3) for nilpotent groups (A. Weiss 91) (Aut) for S n, H S n with H abelian/p-group/s k, for direct products of satisfying groups (ZC2) for some simple groups (PSL(2, p), most sporadic simple groups), but unknown for A n Counterexamples to (ZC2) were first found by Roggenkamp and Scott ( 87) with G = Later Hertweck and Blanchard ( 99) found one of order 96. (ZC1) remains open in general and is for this reason today called The Zassenhaus Conjecture.

17 Results on (ZC1) There has been a lot of work on the Zassenhaus Conjecture, namely it is known to hold for Nilpotent groups (Weiss 91) Groups having a normal Sylow subgroup with an abelian complement (Hertweck 06) Cyclic-by-abelian groups ( 8 papers 83-06; Hertweck 08 and Caicedo, del Río, M. 12) Some families of metabelian groups, not necessarily cyclic-by-abelian (Sehgal, Weiss 86; Marciniak, Ritter, Sehgal, Weiss 87) Some concrete groups (e.g. A 5 (Luthar, Passi 89), S 5 (Luthar, Trama 91), SL(2, 5) (Dokuchaev, Juriaans, Milies 97), groups till order 71 (Hoefert, Kimmerle 06), central extensions of S 5 (Bovdi, Hertweck 08), PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 07), A 6 = PSL(2, 9) (Hertweck 08))

18 Results on (ZC1) There has been a lot of work on the Zassenhaus Conjecture, namely it is known to hold for Nilpotent groups (Weiss 91) Groups having a normal Sylow subgroup with an abelian complement (Hertweck 06) Cyclic-by-abelian groups ( 8 papers 83-06; Hertweck 08 and Caicedo, del Río, M. 12) Some families of metabelian groups, not necessarily cyclic-by-abelian (Sehgal, Weiss 86; Marciniak, Ritter, Sehgal, Weiss 87) Some concrete groups (e.g. A 5 (Luthar, Passi 89), S 5 (Luthar, Trama 91), SL(2, 5) (Dokuchaev, Juriaans, Milies 97), groups till order 71 (Hoefert, Kimmerle 06), central extensions of S 5 (Bovdi, Hertweck 08), PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 07), A 6 = PSL(2, 9) (Hertweck 08))

19 Results on (ZC1) There has been a lot of work on the Zassenhaus Conjecture, namely it is known to hold for Nilpotent groups (Weiss 91) Groups having a normal Sylow subgroup with an abelian complement (Hertweck 06) Cyclic-by-abelian groups ( 8 papers 83-06; Hertweck 08 and Caicedo, del Río, M. 12) Some families of metabelian groups, not necessarily cyclic-by-abelian (Sehgal, Weiss 86; Marciniak, Ritter, Sehgal, Weiss 87) Some concrete groups (e.g. A 5 (Luthar, Passi 89), S 5 (Luthar, Trama 91), SL(2, 5) (Dokuchaev, Juriaans, Milies 97), groups till order 71 (Hoefert, Kimmerle 06), central extensions of S 5 (Bovdi, Hertweck 08), PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 07), A 6 = PSL(2, 9) (Hertweck 08))

20 Results on (ZC1) There has been a lot of work on the Zassenhaus Conjecture, namely it is known to hold for Nilpotent groups (Weiss 91) Groups having a normal Sylow subgroup with an abelian complement (Hertweck 06) Cyclic-by-abelian groups ( 8 papers 83-06; Hertweck 08 and Caicedo, del Río, M. 12) Some families of metabelian groups, not necessarily cyclic-by-abelian (Sehgal, Weiss 86; Marciniak, Ritter, Sehgal, Weiss 87) Some concrete groups (e.g. A 5 (Luthar, Passi 89), S 5 (Luthar, Trama 91), SL(2, 5) (Dokuchaev, Juriaans, Milies 97), groups till order 71 (Hoefert, Kimmerle 06), central extensions of S 5 (Bovdi, Hertweck 08), PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 07), A 6 = PSL(2, 9) (Hertweck 08))

21 Results on (ZC1) There has been a lot of work on the Zassenhaus Conjecture, namely it is known to hold for Nilpotent groups (Weiss 91) Groups having a normal Sylow subgroup with an abelian complement (Hertweck 06) Cyclic-by-abelian groups ( 8 papers 83-06; Hertweck 08 and Caicedo, del Río, M. 12) Some families of metabelian groups, not necessarily cyclic-by-abelian (Sehgal, Weiss 86; Marciniak, Ritter, Sehgal, Weiss 87) Some concrete groups (e.g. A 5 (Luthar, Passi 89), S 5 (Luthar, Trama 91), SL(2, 5) (Dokuchaev, Juriaans, Milies 97), groups till order 71 (Hoefert, Kimmerle 06), central extensions of S 5 (Bovdi, Hertweck 08), PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 07), A 6 = PSL(2, 9) (Hertweck 08))

22 Weaker versions of (ZC1) As the Zassenhaus Conjecture is very hard to prove, sometimes weaker versions are considered. Namely: Isomorphism Problem cyclic: U V(ZG), U finite cyclic. Is U isomorphic to a subgroup of G? Yes, for solvable groups (Hertweck 08). Prime graph question: Let p and q be different primes. If V(ZG) has an element of order pq, does G have an element of this order? (I.e. do G and V(ZG) have the same prime graph?) Yes for G a Frobenius group (Kimmerle 06), many sporadic simple groups (Konovalov, Bovdi et al ), hopefully groups whose order is divisible by at most three primes (Konovalov, Kimmerle 12 and Baechle, M. 12).

23 Weaker versions of (ZC1) As the Zassenhaus Conjecture is very hard to prove, sometimes weaker versions are considered. Namely: Isomorphism Problem cyclic: U V(ZG), U finite cyclic. Is U isomorphic to a subgroup of G? Yes, for solvable groups (Hertweck 08). Prime graph question: Let p and q be different primes. If V(ZG) has an element of order pq, does G have an element of this order? (I.e. do G and V(ZG) have the same prime graph?) Yes for G a Frobenius group (Kimmerle 06), many sporadic simple groups (Konovalov, Bovdi et al ), hopefully groups whose order is divisible by at most three primes (Konovalov, Kimmerle 12 and Baechle, M. 12).

24 Weaker versions of (ZC1) As the Zassenhaus Conjecture is very hard to prove, sometimes weaker versions are considered. Namely: Isomorphism Problem cyclic: U V(ZG), U finite cyclic. Is U isomorphic to a subgroup of G? Yes, for solvable groups (Hertweck 08). Prime graph question: Let p and q be different primes. If V(ZG) has an element of order pq, does G have an element of this order? (I.e. do G and V(ZG) have the same prime graph?) Yes for G a Frobenius group (Kimmerle 06), many sporadic simple groups (Konovalov, Bovdi et al ), hopefully groups whose order is divisible by at most three primes (Konovalov, Kimmerle 12 and Baechle, M. 12).

25 The Cyclic-By-Abelian Case Theorem (Hertweck 08) Let G = AX with A G cyclic and X abelian. Then (ZC1) holds for G. This proved especially the metacyclic case, which had been open till then. Relying on this results and developing the methods further we got: Theorem (Caicedo, M, del Río 12) Let G be cyclic-by-abelian, i.e. G has a cyclic normal subgroup A s.t. G/A is abelian. Then (ZC1) holds for G. Main obstacle in generalizing Hertwecks proof: C G (A) AZ(G).

26 The Cyclic-By-Abelian Case Theorem (Hertweck 08) Let G = AX with A G cyclic and X abelian. Then (ZC1) holds for G. This proved especially the metacyclic case, which had been open till then. Relying on this results and developing the methods further we got: Theorem (Caicedo, M, del Río 12) Let G be cyclic-by-abelian, i.e. G has a cyclic normal subgroup A s.t. G/A is abelian. Then (ZC1) holds for G. Main obstacle in generalizing Hertwecks proof: C G (A) AZ(G).

27 Partial augmentation General difficulty: No constructions for torsion units available. Definition Let u = z g g be a torsion unit in ZG and denote by g G the g G conjugacy class of g in G. Then ε g (u) = z x is called x g G partial augmentation of u in respect to g. Lemma For a torsion unit u V(ZG) there exists an x U(QG) s.t. x 1 ux G if and only if ε g (u k ) 0 for all g G and k N. Study ZC1 by studying partial augmentations.

28 Partial augmentation General difficulty: No constructions for torsion units available. Definition Let u = z g g be a torsion unit in ZG and denote by g G the g G conjugacy class of g in G. Then ε g (u) = z x is called x g G partial augmentation of u in respect to g. Lemma For a torsion unit u V(ZG) there exists an x U(QG) s.t. x 1 ux G if and only if ε g (u k ) 0 for all g G and k N. Study ZC1 by studying partial augmentations.

29 General knowledge on partial augmentations Lemma (Berman, Higman) u = z g g, u 1 a torsion unit in ZG, then z 1 = ε 1 (u) = 0. g G Lemma (Marciniak, Ritter, Sehgal, Weiss; Hertweck) Let u V(ZG) be of finite order. If ε g (u) 0, then the order of g divides the order of u. Lemma (Hertweck) For u V(ZG) suppose that the p-part of u is conjugate in Z p G to an element x G. Then ε g (u) = 0, if the p-part of g is not conjugate to x. Here Z p denotes the p-adic integers.

30 General knowledge on partial augmentations Lemma (Berman, Higman) u = z g g, u 1 a torsion unit in ZG, then z 1 = ε 1 (u) = 0. g G Lemma (Marciniak, Ritter, Sehgal, Weiss; Hertweck) Let u V(ZG) be of finite order. If ε g (u) 0, then the order of g divides the order of u. Lemma (Hertweck) For u V(ZG) suppose that the p-part of u is conjugate in Z p G to an element x G. Then ε g (u) = 0, if the p-part of g is not conjugate to x. Here Z p denotes the p-adic integers.

31 General knowledge on partial augmentations Lemma (Berman, Higman) u = z g g, u 1 a torsion unit in ZG, then z 1 = ε 1 (u) = 0. g G Lemma (Marciniak, Ritter, Sehgal, Weiss; Hertweck) Let u V(ZG) be of finite order. If ε g (u) 0, then the order of g divides the order of u. Lemma (Hertweck) For u V(ZG) suppose that the p-part of u is conjugate in Z p G to an element x G. Then ε g (u) = 0, if the p-part of g is not conjugate to x. Here Z p denotes the p-adic integers.

32 Proof strategy Inductive approach: Assume u V(ZG) is a minimal counterexample to ZC, i.e every proper power of u is rationally conjugate to a group element and ZC holds for proper subgroups and quotients of G. Then a group theoretic observation yields Theorem (del Río, Sehgal 06) Let A be an abelian normal subgroup of G with abelian quotient. Then ε g (u) 0 for g G\C G (A). Similarly one gets: Lemma Let A G, A cyclic, G/A abelian and set D = Z(C G (A)). Then ε g (u) 0 for g G\ D. Study partial augmentations only in D = Z(C G (A)).

33 Proof strategy Inductive approach: Assume u V(ZG) is a minimal counterexample to ZC, i.e every proper power of u is rationally conjugate to a group element and ZC holds for proper subgroups and quotients of G. Then a group theoretic observation yields Theorem (del Río, Sehgal 06) Let A be an abelian normal subgroup of G with abelian quotient. Then ε g (u) 0 for g G\C G (A). Similarly one gets: Lemma Let A G, A cyclic, G/A abelian and set D = Z(C G (A)). Then ε g (u) 0 for g G\ D. Study partial augmentations only in D = Z(C G (A)).

34 Two cases For a normal subgroup N of G denote by ω N : ZG Z(G/N) the linear extension of the natural homomorphism G G/N. Let A G, A cyclic, G/A abelian, D = Z(C G (A)) and assume u is a minimal counterexample to ZC. Then we study separately: ω D (u) = 1. Using p-adic methods, especially Weiss double action formalism and "Permutation module"-results and Cliff-Weiss Theorem (details below). ω D (u) 1. Using the Luthar-Passi method, which relates eigenvalues under complex representations and partial augmentations (not in this talk).

35 Two cases For a normal subgroup N of G denote by ω N : ZG Z(G/N) the linear extension of the natural homomorphism G G/N. Let A G, A cyclic, G/A abelian, D = Z(C G (A)) and assume u is a minimal counterexample to ZC. Then we study separately: ω D (u) = 1. Using p-adic methods, especially Weiss double action formalism and "Permutation module"-results and Cliff-Weiss Theorem (details below). ω D (u) 1. Using the Luthar-Passi method, which relates eigenvalues under complex representations and partial augmentations (not in this talk).

36 Two cases For a normal subgroup N of G denote by ω N : ZG Z(G/N) the linear extension of the natural homomorphism G G/N. Let A G, A cyclic, G/A abelian, D = Z(C G (A)) and assume u is a minimal counterexample to ZC. Then we study separately: ω D (u) = 1. Using p-adic methods, especially Weiss double action formalism and "Permutation module"-results and Cliff-Weiss Theorem (details below). ω D (u) 1. Using the Luthar-Passi method, which relates eigenvalues under complex representations and partial augmentations (not in this talk).

37 Translating conjugacy questions into module questions Weiss double action formalism: For a finite group H let α,β : H V(ZG) be homomorphisms. Then α and β are called Q-equivalent if there exists x U(QG) s.t. x 1 α(h)x = β(h) for all h H. ZG becomes a Z(G H)-module via m (g, h) = g 1 mα(h), denoted (ZG) α. We have Lemma a) α and β are Q-equivalent iff (ZG) α = (ZG)β. b) α is Q-equivalent to a homomorphism H G (followed by inclusion G V(ZG)) iff (ZG) α is a permutation lattice for G H, i.e. there exist U 1,..., U n G H s.t. (ZG) α n = indu G H i 1. i=1 c) The character of (ZG) α is χ(g, h) = C g (G) ε g (α(h)).

38 Translating conjugacy questions into module questions Weiss double action formalism: For a finite group H let α,β : H V(ZG) be homomorphisms. Then α and β are called Q-equivalent if there exists x U(QG) s.t. x 1 α(h)x = β(h) for all h H. ZG becomes a Z(G H)-module via m (g, h) = g 1 mα(h), denoted (ZG) α. We have Lemma a) α and β are Q-equivalent iff (ZG) α = (ZG)β. b) α is Q-equivalent to a homomorphism H G (followed by inclusion G V(ZG)) iff (ZG) α is a permutation lattice for G H, i.e. there exist U 1,..., U n G H s.t. (ZG) α n = indu G H i 1. i=1 c) The character of (ZG) α is χ(g, h) = C g (G) ε g (α(h)).

39 The matrix method of Marciniak, Ritter, Sehgal, Weiss For N G set k = [G : N] and view ZG as ZG = k ZNg i = (ZN) k. Then the multiplication with an i=1 element from V(ZG) defines a matrix in GL k (ZN). This way a homomorphism α : H V(ZG) becomes a homomorphism H GL k (ZN) and (ZG) α becomes a Z(GL k (ZN) GL k (ZN)) -module. For u V(ZG) and g G of the same order let A u and A g define the corresponding "multiplication-matrix" on (ZN) k. Set c = u and α : c A u, β : c A g. Then u and g are rationally conjugate if and only if (ZG) α = (ZG)β. And ε n (u) 0 n N if and only if A X u = A g with X GL k (QN). Study rational conjugation in V(ZG) by studying conjugacy classes of elements of finite order in GL k (ZN).

40 The matrix method of Marciniak, Ritter, Sehgal, Weiss For N G set k = [G : N] and view ZG as ZG = k ZNg i = (ZN) k. Then the multiplication with an i=1 element from V(ZG) defines a matrix in GL k (ZN). This way a homomorphism α : H V(ZG) becomes a homomorphism H GL k (ZN) and (ZG) α becomes a Z(GL k (ZN) GL k (ZN)) -module. For u V(ZG) and g G of the same order let A u and A g define the corresponding "multiplication-matrix" on (ZN) k. Set c = u and α : c A u, β : c A g. Then u and g are rationally conjugate if and only if (ZG) α = (ZG)β. And ε n (u) 0 n N if and only if A X u = A g with X GL k (QN). Study rational conjugation in V(ZG) by studying conjugacy classes of elements of finite order in GL k (ZN).

41 Cliff-Weiss Theorem Let ε k : GL k (ZN) GL k (Z) be the augmentation map applied on every entry and set ker(ε k ) = SGL k (ZN). Now let N G, [G : N] = k and let u V(ZG) be a torsion unit. Then under the homomorphism u GL k (ZN) described above α(u) is an element of SGL k (ZN) if and only if ω N (u) = 1. A stronger version of the Zassenhaus Conjecture Is every matrix A SGL k (ZN) of finite order conjugate to a diagonal matrix with entries in N in the group GL k (QN)? (For k = 1 this is ZC.) Theorem (Cliff, Weiss 00) Let N be a nilpotent group. Then every matrix of finite order in SGL k (ZN) is conjugate in GL k (QN) to a diagonal matrix with entries in N for every k if and only if N has at most one non-cyclic Sylow subgroup.

42 Cliff-Weiss Theorem Let ε k : GL k (ZN) GL k (Z) be the augmentation map applied on every entry and set ker(ε k ) = SGL k (ZN). Now let N G, [G : N] = k and let u V(ZG) be a torsion unit. Then under the homomorphism u GL k (ZN) described above α(u) is an element of SGL k (ZN) if and only if ω N (u) = 1. A stronger version of the Zassenhaus Conjecture Is every matrix A SGL k (ZN) of finite order conjugate to a diagonal matrix with entries in N in the group GL k (QN)? (For k = 1 this is ZC.) Theorem (Cliff, Weiss 00) Let N be a nilpotent group. Then every matrix of finite order in SGL k (ZN) is conjugate in GL k (QN) to a diagonal matrix with entries in N for every k if and only if N has at most one non-cyclic Sylow subgroup.

43 Cliff-Weiss Theorem Let ε k : GL k (ZN) GL k (Z) be the augmentation map applied on every entry and set ker(ε k ) = SGL k (ZN). Now let N G, [G : N] = k and let u V(ZG) be a torsion unit. Then under the homomorphism u GL k (ZN) described above α(u) is an element of SGL k (ZN) if and only if ω N (u) = 1. A stronger version of the Zassenhaus Conjecture Is every matrix A SGL k (ZN) of finite order conjugate to a diagonal matrix with entries in N in the group GL k (QN)? (For k = 1 this is ZC.) Theorem (Cliff, Weiss 00) Let N be a nilpotent group. Then every matrix of finite order in SGL k (ZN) is conjugate in GL k (QN) to a diagonal matrix with entries in N for every k if and only if N has at most one non-cyclic Sylow subgroup.

44 The proof Hertweck can reduce the case ω CG (A)(u) = 1 to the case ω A (u) = 1, where Cliff-Weiss is available, since A is cyclic. We use D = Z(C G (A)) instead and this is in general not cyclic. Lemma (extracted out of Cliff and Weiss paper) Let N be an abelian normal subgroup of G, u a torsion unit in ZG satisfying ω N (u) = 1, η an irreducible character of N and n N. Then C G (hn) : N ε G hn (u) 0. h kerη Combining this with the following Theorem analogues to a Theorem of Hertweck, but with a more technical proof, we obtain our result. (Z p denotes the p-adic integers.) Theorem If the order of u is the power of a prime p, then u is conjugate in Z p G to an element of D p.

45 The proof Hertweck can reduce the case ω CG (A)(u) = 1 to the case ω A (u) = 1, where Cliff-Weiss is available, since A is cyclic. We use D = Z(C G (A)) instead and this is in general not cyclic. Lemma (extracted out of Cliff and Weiss paper) Let N be an abelian normal subgroup of G, u a torsion unit in ZG satisfying ω N (u) = 1, η an irreducible character of N and n N. Then C G (hn) : N ε G hn (u) 0. h kerη Combining this with the following Theorem analogues to a Theorem of Hertweck, but with a more technical proof, we obtain our result. (Z p denotes the p-adic integers.) Theorem If the order of u is the power of a prime p, then u is conjugate in Z p G to an element of D p.

46 The proof Hertweck can reduce the case ω CG (A)(u) = 1 to the case ω A (u) = 1, where Cliff-Weiss is available, since A is cyclic. We use D = Z(C G (A)) instead and this is in general not cyclic. Lemma (extracted out of Cliff and Weiss paper) Let N be an abelian normal subgroup of G, u a torsion unit in ZG satisfying ω N (u) = 1, η an irreducible character of N and n N. Then C G (hn) : N ε G hn (u) 0. h kerη Combining this with the following Theorem analogues to a Theorem of Hertweck, but with a more technical proof, we obtain our result. (Z p denotes the p-adic integers.) Theorem If the order of u is the power of a prime p, then u is conjugate in Z p G to an element of D p.

47 Thank You! Thank you for your attention! Enjoy the conference dinner!