Coleman automorphisms of finite groups and their minimal normal subgroups

Abstract

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Lastly, we note that class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge.

Coleman automorphisms of finite groups with a self-centralizing normal subgroup

“…As an immediate consequence of Corollary 1.1, we have: P r o o f. This is a direct consequence of Theorem 3.2 in [19].…”

“…In this case, the group N is a normal q-subgroup of G for some prime q ∈ π(G). By Lemma 2.4, C G (N ) N , thus the assertion follows from Theorem 2.2 in [19].…”

“…For instance, Hertweck and Kimmerle in [8] proved that Out Col (G) = 1 for a quasinilpotent group G. It follows that the normalizer property holds for finite quasinilpotent groups. Van Antwerpen in [19] proved that all Coleman automorphisms of a finite group with a self-central normal p-subgroup are inner. Petit Lobão and Sehgal in [16] showed that the normalizer property holds for the wreath product G = N wrS m of a finite nilpotent group N by symmetric group S m .…”

Coleman Automorphisms of Some Metabelian Groups

“…In [2], Krempa proved that Out Z (F) is a 2-group. us, if we show that Out Col (F) is a group of odd order or Aut Col (F) Inn(F) under some conditions, then Aut Z (F) Inn(F), that is, the normalizer problem holds for F. Related results on this subject can be found in [3][4][5][6][7][8][9]. e purpose of this study is to determine the structure of the Coleman automorphism groups of some metabelian groups.…”

The Modular Isomorphism Problem: a survey

“…A counterexample to the Normalizer Problem was the cornerstone of Hertweck's construction of a counterexample to (Z-IP) using a particular unit in V (ZG) violating the condition of the Normalizer Problem to construct a non-isomorphic group basis. Nevertheless, the Normalizer Problem is still of interest, see [VA18]. A very general positive result on the problem was achieved by Jackowski and Marciniak: Theorem 2.9.…”