on coleman automorphisms of finite groups and their ... · normal subgroups arne van antwerpen may...
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071215/6045b84d44c25c0cd837d175/html5/thumbnails/1.jpg)
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ON COLEMAN AUTOMORPHISMS OFFINITE GROUPS AND THEIR MINIMAL
NORMAL SUBGROUPS
Arne Van AntwerpenMay 23, 2017
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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).
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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).
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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).
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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).
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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
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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](https://reader035.vdocuments.net/reader035/viewer/2022071215/6045b84d44c25c0cd837d175/html5/thumbnails/8.jpg)
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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)
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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)
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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)
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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)
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.