Coleman automorphisms of permutational wreath products II

In this paper, we mainly investigate the class-preserving Coleman automorphisms of finite groups whose Sylow 2-subgroups are semidihedral. We prove that if [Formula: see text] is a finite solvable group with semidihedral Sylow 2-subgroups, then [Formula: see text] is a [Formula: see text]-group and therefore [Formula: see text] satisfies the normalizer property. As some applications of this result, we also investigate the normalizer property of the following groups: the groups whose Sylow 2-subgroups are semidihedral and Sylow subgroups of odd order are all cyclic, the groups [Formula: see text] with [Formula: see text] a nilpotent normal subgroup and [Formula: see text] a maximal class 2-group, and the wreath products [Formula: see text] with [Formula: see text] a group whose Sylow 2-subgroups are of maximal class with order [Formula: see text] and [Formula: see text] a rational permutation group.

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Class-preserving Coleman automorphisms of some classes of finite groups

Hai

Ling

2022

Open Mathematics

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The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.

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Coleman automorphisms of permutational wreath products II

Cited by 4 publications

References 22 publications

The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian

The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian

Class-preserving Coleman automorphisms of finite groups whose Sylow 2-subgroups are semidihedral

Class-preserving Coleman automorphisms of some classes of finite groups

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