On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May...

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ON COLEMAN AUTOMORPHISMS OF FINITE GROUPS AND THEIR MINIMAL NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017

Transcript of On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May...

Page 1: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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ON COLEMAN AUTOMORPHISMS OFFINITE GROUPS AND THEIR MINIMAL

NORMAL SUBGROUPS

Arne Van AntwerpenMay 23, 2017

Page 2: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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DEFINITIONS

DefinitionLet G be a finite group and σ ∈ Aut(G). If for any prime p dividingthe order of G and any Sylow p-subgroup P of G, there exists ag ∈ G such that σ|P = conj(g)|P, then σ is said to be a Colemanautomorphism.

Denote Autcol(G) for the set of Coleman automorphisms, Inn(G)for the set of inner automorphisms and set

Outcol(G) = Autcol(G)/Inn(G).

Page 3: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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DEFINITIONS

DefinitionLet G be a finite group and σ ∈ Aut(G). If for any prime p dividingthe order of G and any Sylow p-subgroup P of G, there exists ag ∈ G such that σ|P = conj(g)|P, then σ is said to be a Colemanautomorphism.Denote Autcol(G) for the set of Coleman automorphisms, Inn(G)for the set of inner automorphisms and set

Outcol(G) = Autcol(G)/Inn(G).

Page 4: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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ELEMENTARY RESULTS

Theorem (Hertweck and Kimmerle)Let G be a finite group. The prime divisors of |Autcol(G)| are alsoprime divisors of |G|.

LemmaLet N be a normal subgroup of a finite group G. If σ ∈ Autcol(G),then σ|N ∈ Aut(N).

Page 5: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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ELEMENTARY RESULTS

Theorem (Hertweck and Kimmerle)Let G be a finite group. The prime divisors of |Autcol(G)| are alsoprime divisors of |G|.

LemmaLet N be a normal subgroup of a finite group G. If σ ∈ Autcol(G),then σ|N ∈ Aut(N).

Page 6: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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NORMALIZER PROBLEM

Let G be a group and R a ring (denote U(RG) for the units of RG),then we clearly have that

GCU(RG)(G) ⊆ NU(RG)(G)

Is this an equality?

NU(RG)(G) = GCU(RG)(G)

Of special interest: R = Z

Page 7: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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NORMALIZER PROBLEM

Let G be a group and R a ring (denote U(RG) for the units of RG),then we clearly have that

GCU(RG)(G) ⊆ NU(RG)(G)

Is this an equality?

NU(RG)(G) = GCU(RG)(G)

Of special interest: R = Z

Page 8: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SETUP FOR REFORMULATION

If u ∈ NU(RG)(G), then u induces an automorphism of G:

ϕu : G→ Gg 7→ u−1gu

Denote AutU(G;R) for the group of these automorphisms(AutU(G) if R = Z) and OutU(G;R) = AutU(G;R)/Inn(G)

Page 9: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SETUP FOR REFORMULATION

If u ∈ NU(RG)(G), then u induces an automorphism of G:

ϕu : G→ Gg 7→ u−1gu

Denote AutU(G;R) for the group of these automorphisms(AutU(G) if R = Z) and OutU(G;R) = AutU(G;R)/Inn(G)

Page 10: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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REFORMULATION

Theorem (Jackowski and Marciniak)G a finite group, R a commutative ring. TFAE1. NU(RG)(G) = GCU(RG)(G)

2. AutU(G;R) = Inn(G)

TheoremLet G be a finite group. Then,

AutU(G) ⊆ Autcol(G)

Page 11: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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REFORMULATION

Theorem (Jackowski and Marciniak)G a finite group, R a commutative ring. TFAE1. NU(RG)(G) = GCU(RG)(G)

2. AutU(G;R) = Inn(G)

TheoremLet G be a finite group. Then,

AutU(G) ⊆ Autcol(G)

Page 12: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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MORE KNOWN RESULTS

Lemma (Hertweck)Let p be a prime number and α ∈ Aut(G) of p-power order.Assume that there exists a normal subgroup N of G, such thatα|N = idN and α induces identity on G/N. Then α induces identityon G/Op(Z(N)). Moreover, if α also fixes a Sylow p-subgroup of Gelementwise, i.e. α is p-central, then α is conjugation by anelement of Op(Z(N)).

Theorem (Hertweck and Kimmerle)For any finite simple group G, there is a prime p dividing |G| suchthat p-central automorphisms of G are inner automorphisms.

Page 13: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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MORE KNOWN RESULTS

Lemma (Hertweck)Let p be a prime number and α ∈ Aut(G) of p-power order.Assume that there exists a normal subgroup N of G, such thatα|N = idN and α induces identity on G/N. Then α induces identityon G/Op(Z(N)). Moreover, if α also fixes a Sylow p-subgroup of Gelementwise, i.e. α is p-central, then α is conjugation by anelement of Op(Z(N)).

Theorem (Hertweck and Kimmerle)For any finite simple group G, there is a prime p dividing |G| suchthat p-central automorphisms of G are inner automorphisms.

Page 14: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SELF-CENTRALITY

Theorem (A.V.A.)Let G be a normal subgroup of a finite group K. Let N be aminimal non-trivial characteristic subgroup of G. If CK(N) ⊆ N,then every Coleman automorphism of K is inner. In particular, thenormalizer problem holds for these groups.

Proof.1. N is characteristically simple2. Restrict to simple groups3. Subdivide Abelian vs. Non-abelian4. Hertweck’s result

Page 15: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SELF-CENTRALITY

Theorem (A.V.A.)Let G be a normal subgroup of a finite group K. Let N be aminimal non-trivial characteristic subgroup of G. If CK(N) ⊆ N,then every Coleman automorphism of K is inner. In particular, thenormalizer problem holds for these groups.

Proof.1. N is characteristically simple

2. Restrict to simple groups3. Subdivide Abelian vs. Non-abelian4. Hertweck’s result

Page 16: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SELF-CENTRALITY

Theorem (A.V.A.)Let G be a normal subgroup of a finite group K. Let N be aminimal non-trivial characteristic subgroup of G. If CK(N) ⊆ N,then every Coleman automorphism of K is inner. In particular, thenormalizer problem holds for these groups.

Proof.1. N is characteristically simple2. Restrict to simple groups

3. Subdivide Abelian vs. Non-abelian4. Hertweck’s result

Page 17: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SELF-CENTRALITY

Theorem (A.V.A.)Let G be a normal subgroup of a finite group K. Let N be aminimal non-trivial characteristic subgroup of G. If CK(N) ⊆ N,then every Coleman automorphism of K is inner. In particular, thenormalizer problem holds for these groups.

Proof.1. N is characteristically simple2. Restrict to simple groups3. Subdivide Abelian vs. Non-abelian

4. Hertweck’s result

Page 18: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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SELF-CENTRALITY

Theorem (A.V.A.)Let G be a normal subgroup of a finite group K. Let N be aminimal non-trivial characteristic subgroup of G. If CK(N) ⊆ N,then every Coleman automorphism of K is inner. In particular, thenormalizer problem holds for these groups.

Proof.1. N is characteristically simple2. Restrict to simple groups3. Subdivide Abelian vs. Non-abelian4. Hertweck’s result

Page 19: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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COROLLARY 1

CorollaryLet G = Po H be a semidirect product of a finite p-group P and afinite group H. If CG(P) ⊆ P, then G has no non-inner Colemanautomorphisms.

Page 20: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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COROLLARY 2

The wreath product G oH of G,H is defined as Πh∈HGoH, whereH acts on the indices.

CorollaryLet S be a finite simple group, I a finite set of indices, and H anyfinite group. If G = Πi∈ISo H is a group such thatCG(Πi∈IS) ⊆ Z(Πi∈IS). Then G has no non-inner Colemanautomorphism. In particular, the normalizer problem holds for G.Then, in particular, the wreath product S o H has no non-innerColeman automorphisms. Moreover, in both cases thenormalizer problem holds for G.

Page 21: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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COROLLARY 2

The wreath product G oH of G,H is defined as Πh∈HGoH, whereH acts on the indices.

CorollaryLet S be a finite simple group, I a finite set of indices, and H anyfinite group. If G = Πi∈ISo H is a group such thatCG(Πi∈IS) ⊆ Z(Πi∈IS). Then G has no non-inner Colemanautomorphism. In particular, the normalizer problem holds for G.Then, in particular, the wreath product S o H has no non-innerColeman automorphisms. Moreover, in both cases thenormalizer problem holds for G.

Page 22: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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QUESTIONS OF HERTWECK AND KIMMERLE

Questions (Hertweck and Kimmerle)

1. Is Outcol(G) a p′-group if G does not have Cp as a chieffactor?

2. Is Outcol(G) trivial if Op′(G) is trivial?3. Is Outcol(G) trivial if G has a unique minimal non-trivial

normal subgroup?

Partial Answer (Hertweck, Kimmerle)Besides giving several conditions, all three statements hold if Gis assumed to be a p-constrained group.

Page 23: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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QUESTIONS OF HERTWECK AND KIMMERLE

Questions (Hertweck and Kimmerle)

1. Is Outcol(G) a p′-group if G does not have Cp as a chieffactor?

2. Is Outcol(G) trivial if Op′(G) is trivial?

3. Is Outcol(G) trivial if G has a unique minimal non-trivialnormal subgroup?

Partial Answer (Hertweck, Kimmerle)Besides giving several conditions, all three statements hold if Gis assumed to be a p-constrained group.

Page 24: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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QUESTIONS OF HERTWECK AND KIMMERLE

Questions (Hertweck and Kimmerle)

1. Is Outcol(G) a p′-group if G does not have Cp as a chieffactor?

2. Is Outcol(G) trivial if Op′(G) is trivial?3. Is Outcol(G) trivial if G has a unique minimal non-trivial

normal subgroup?

Partial Answer (Hertweck, Kimmerle)Besides giving several conditions, all three statements hold if Gis assumed to be a p-constrained group.

Page 25: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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QUESTIONS OF HERTWECK AND KIMMERLE

Questions (Hertweck and Kimmerle)

1. Is Outcol(G) a p′-group if G does not have Cp as a chieffactor?

2. Is Outcol(G) trivial if Op′(G) is trivial?3. Is Outcol(G) trivial if G has a unique minimal non-trivial

normal subgroup?

Partial Answer (Hertweck, Kimmerle)Besides giving several conditions, all three statements hold if Gis assumed to be a p-constrained group.

Page 26: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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NEW ANSWERS

Partial Answers (Van Antwerpen)

1. No new result.

2. True, if Op(G) = Op′(G) = 1, where p is an odd prime and theorder of every direct component of E(G) is divisible by p.

3. True, if the unique minimal non-trivial normal subgroup isnon-abelian. True, if question 2 has a positive answer.

Page 27: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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NEW ANSWERS

Partial Answers (Van Antwerpen)

1. No new result.2. True, if Op(G) = Op′(G) = 1, where p is an odd prime and the

order of every direct component of E(G) is divisible by p.

3. True, if the unique minimal non-trivial normal subgroup isnon-abelian. True, if question 2 has a positive answer.

Page 28: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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NEW ANSWERS

Partial Answers (Van Antwerpen)

1. No new result.2. True, if Op(G) = Op′(G) = 1, where p is an odd prime and the

order of every direct component of E(G) is divisible by p.3. True, if the unique minimal non-trivial normal subgroup is

non-abelian. True, if question 2 has a positive answer.

Page 29: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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FURTHER RESEARCH

Inkling of IdeaIn case G has a unique minimal non-trivial normal subgroup N,which we may assume to be abelian, we may be able to use theclassification of the simple groups and the list of all Schurmultipliers to give a technical proof.

Related QuestionGross conjectured that for a finite group G with Op′(G) = 1 forsome odd prime p, p-central automorphisms of p-power orderare inner. Hertweck and Kimmerle believed this is possible usingthe classification of Schur mutipliers.

Page 30: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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FURTHER RESEARCH

Inkling of IdeaIn case G has a unique minimal non-trivial normal subgroup N,which we may assume to be abelian, we may be able to use theclassification of the simple groups and the list of all Schurmultipliers to give a technical proof.

Related QuestionGross conjectured that for a finite group G with Op′(G) = 1 forsome odd prime p, p-central automorphisms of p-power orderare inner. Hertweck and Kimmerle believed this is possible usingthe classification of Schur mutipliers.

Page 31: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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FURTHER RESEARCH 2

Stretch of IdeaM. Murai connected Coleman automorphisms of finite groups tothe theory of blocks in representation theory, showing severalslight generalizations of known theorems. Looking into hismethods may prove useful.

Page 32: On Coleman automorphisms of finite groups and their ... · NORMAL SUBGROUPS Arne Van Antwerpen May 23, 2017. 2 DEFINITIONS De˙nition Let G be a ˙nite group and ˙2Aut(G). ... Let

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REFERENCES

1. M. Hertweck and W. Kimmerle. Coleman automorphismsof finite groups.

2. S.Jackowski and Z.Marciniak. Group automorphismsinducing the identity map on cohomology.

3. F. Gross. Automorphisms which centralize a Sylowp-subgroup.

4. E.C. Dade. Locally trivial outer automorphisms of finitegroups.

5. J. Hai and J. Guo. The normalizer property for the integralgroup ring of the wreath product of two symmetric groupsSk and Sn.