Coleman automorphisms of finite groups

Abstract

An automorphism σ of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer N of G in the units U of the integral group ZG. Let Out Col (G) be the image of these automorphisms in Out(G). We prove that Out Col (G) is always an abelian group (based on previous work of E. C. Dade, who showed that Out Col (G) is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out Col (G) is a p -group. If O p (G) = 1, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of G/O 2 (G) has order 2, then N = GZ, where Z is the center of U. This improves an earlier result of S.

On Coleman automorphisms of wreath products of finite nilpotent groups by abelian groups

“…Let us remark that the same notion "Coleman automorphisms" also occurred in [8], which has different meanings with that mentioned above. Coleman automorphisms of a finite group G introduced in [8] are defined to be the automorphisms whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G. So Coleman automorphisms introduced in [14] are definitely that introduced in [8], but the converse may be false (see [2,Example 2.11] and [7, Example 2.1]). Coleman automorphisms discussed in this paper always refer to that introduced by Marciniak and Roggenkamp [14].…”

Coleman Automorphisms of Permutational Wreath Products

“…Let be an automorphism of a group G. Recall that is called a Coleman automorphism if the restriction of to any Sylow subgroup of G equals the restriction of some inner automorphism of G. This definition was initially introduced by Hertweck and Kimmerle in [7]. Denote by Aut Col G the group of all Coleman automorphisms and Inn G the inner automorphism group of G, respectively.…”

Coleman automorphisms of generalized dihedral groups

“…Instead, we shall focus on determining the structure of Coleman automorphism groups in this paper. Dade [2] proved that Out Col (G) is nilpotent for any group G. Hertweck and Kimmerle [4] improved Dade's result and proved that Out Col (G) is actually abelian for any group G. There are explicit characterizations on Coleman automorphism groups, but most of the time they state that Out Col (G) = 1.…”

“…Thus it is no surprise that Coleman automorphisms of G occur naturally in the study of the normalizer problem. Related work in this direction can be found in [3][4][5][6][7][8][9][10][11][12]. However, we do not intend to go into details for this.…”