Class preserving automorphisms of finite <i>p</i>-groups: a survey

Yadav, Manoj K.

doi:10.1017/cbo9780511842474.019

Cited by 15 publications

(18 citation statements)

References 27 publications

(41 reference statements)

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“…Notice that Inn(G), the group of all inner automorphisms of G, is a normal subgroup of Aut c (G) and Aut c (G) acts trivially on the center of G. A detailed survey on class-preserving automorphisms of finite p-groups can be found in [25].…”

Section: Proof Of Theoremmentioning

confidence: 99%

Central Quotient Versus Commutator Subgroup of Groups

Yadav

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Section: Proof Of Theoremmentioning

confidence: 99%

Central Quotient Versus Commutator Subgroup of Groups

Yadav

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…More groups with non-inner class-preserving automorphism were constructed in [2,11,13,21,28]. We refer the reader to [30] for a more comprehensive survey on the topic. So Theorem A above classifies all groups of order p 6 , p an odd prime, having non-inner class-preserving automorphisms.…”

Section: Moreovermentioning

confidence: 99%

“…Theorem A. Let G be a group of order p 6 for an odd prime p. Then Out c (G) = 1 if and only if G belongs to one of the isoclinism families Φ k for k = 7, 10, 13,15,18,20,21,24,30,36,38,39.…”

Section: Introductionmentioning

confidence: 99%

On Ш-rigidity of groups of orderp6

Pradeep

Yadav

2015

Journal of Algebra

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“…Examples of groups of larger nilpotency class and satisfying Hypothesis A are given below. A (bit wild) natural problem which arises here [21,Problem 6.7] is the following:…”

Section: Introductionmentioning

confidence: 99%

Class-preserving automorphisms of finite p-groups II

Yadav

2015

Isr. J. Math.

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Let G be a finite group minimally generated by d(G) elements and Autc(G) denote the group of all (conjugacy) class-preserving automorphisms of G. Continuing our work [Class preserving automorphisms of finite p-groups, J. London Math. Soc. 75(3) (2007), 755-772], we study finite p-groups G such that | Autc(G)| = |γ 2 (G)| d(G) , where γ 2 (G) denotes the commutator subgroup of G. If G is such a p-group of class 2, then we show that d(G) is even, 2d(γ 2 (G)) ≤ d(G) and G/ Z(G) is homocyclic. When the nilpotency class of G is larger than 2, we obtain the following (surprising) results: G) if and only if G is a 2-generator group with cyclic commutator subgroup, where γ 3 (G) denotes the third term in the lower central G) if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian commutator subgroup of order at most 8. As an application, we classify finite nilpotent groups G such that the central quotient G/ Z(G) of G by it's center Z(G) is of the largest possible order. For proving these results, we introduce a generalization of Camina groups and obtain some interesting results. We use Lie theoretic techniques and computer algebra system 'Magma' as tools.where [x, G] denotes the set {[x, g] | g ∈ G} and γ 2 (G) denotes the commutator subgroup of G. We say that a finite group G, minimally generated by d elements, satisfies Hypothesis A if equality holds for it in (1.2). Most obvious examples of groups satisfying Hypothesis A are abelian groups, and little less obvious ones being finite extraspecial p-groups. Notice that none of these classes of groups admit any class preserving outer automorphism.An interesting class of groups G satisfying Hypothesis A was constructed by Burnside [6] (in 1913) while answering his own question [5, page 463] about the existence of a finite group admitting a non-inner class-preserving automorphism. This group is of order p 6 and is isomorphic 2010 Mathematics Subject Classification. Primary 20D15, 20D45.

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Class preserving automorphisms of finite p-groups: a survey

Abstract: In this short survey article, we try to list maximum number of known results on class preserving automorphisms of finite p-groups.

Cited by 15 publications

References 27 publications

Central Quotient Versus Commutator Subgroup of Groups

Central Quotient Versus Commutator Subgroup of Groups

On Ш-rigidity of groups of orderp6

Class-preserving automorphisms of finite p-groups II

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