Class-preserving automorphisms and the normalizer property for Blackburn groups

Abstract

Abstract. For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if N U ðGÞ ¼ ZðUÞG. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which classpreserving automorphisms are inner automorphisms, Out c ðGÞ ¼ 1, have the normalizer property. Recently, Herman and Li have shown that Out c ðGÞ ¼ 1 for a finite Blackburn group G. We show that Out c ðGÞ ¼ 1 for the members G of certain classes of metabelian groups, from which the Herman-Li result follows.Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group Z y ðUÞ of hypercentral units of U is contained in ZðUÞG.

Some results on C-automorphisms of finite nilpotent by nilpotent groups

“…In particular, the normalizer property holds for finite groups of odd order. Other positive results on this problem can also be found in [4][5][6][7][8][9][10][11][12][13][14][15]. Denoted by Aut Z (G) the group consists of all the automorphisms of G inducing an inner automorphism of ZG.…”

Coleman automorphisms of finite groups and their minimal normal subgroups

“…The first results for p-groups, nilpotent groups and groups with normal Sylow 2-subgroups are due (respectively) to D. Coleman [2], A. Saksonov [21], S. Jackowski and Z. Marciniak [16]. Further work by, among others, S. Sehgal, M. Parmenter, Y. Li [1], E. Jespers and M. Hertweck [13], M. Hertweck [11,12,9] and M. Hertweck and W. Kimmerle [14] enlarged the class of groups with a positive answer greatly. M. Mazur [19,20] showed that the question is closely related to the Isomorphism problem.…”

Coleman Automorphisms of Permutational Wreath Products

“…Hertweck [2] constructed a group G of order 2 25 · 97 2 for which the normalizer property fails. For positive results on the normalizer problem, the reader may refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. According to the Coleman's result ([7, Coleman Lemma]), the normalizer property holds for G provided that Out Col G = 1.…”