Class-preserving automorphisms of finite p-groups II

Abstract

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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“…We would like to remark that Theorem 3.9 is also obtained in [26,Theorem 11.2] as a consequence on study of class-preserving automorphisms of finite p-group. But we have presented here a direct proof.…”

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