Class-preserving Coleman automorphisms of some classes of finite groups

Abstract

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 GG be a finite group with a nilpotent subgroup NN. Suppose that G/NG\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of NN. Then it is proved that every class-preserving Coleman automorphism of GG is an inner automorphism. In addition, if GG is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of GG 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.

Keywords

• class-preserving automorphisms
• coleman automorphisms
• the normalizer property
• 20c05
• 20e36
• 16s34