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Class-preserving automorphisms of finite groups. (English) Zbl 0993.20017
An automorphism of a finite group $$G$$ mapping each group element to some conjugate of it is said to be a class-preserving automorphism. The author of this paper is interested in classifying the A-groups (i.e., soluble groups whose Sylow subgroups are Abelian) which are in a certain sense “minimal” with respect to having class-preserving, non-inner automorphisms.
He classifies the A-groups with class-preserving, non-inner automorphisms such that class-preserving automorphisms of proper subgroups and factor groups are inner automorphisms as some subgroups of affine semi-linear groups (Theorem A). These groups are described in Theorem A’: They are Frobenius groups and the group of class-preserving automorphisms modulo the inner ones has order $$p$$ (Theorem A’). In particular, A-groups with class-preserving non-inner automorphisms have a section isomorphic to a group described in Theorem A’ (Corollary 1). Another consequence is that in the case of A-groups with elementary Abelian Sylow subgroups, all class-preserving automorphisms are inner automorphisms (Corollary 2).
He also proves that for a finite group $$G$$ with a nilpotent normal subgroup $$N$$ such that $$G/N$$ is nilpotent and $$G$$ has Abelian Sylow $$p$$-subgroups for a prime $$p$$, any automorphism of $$G$$ of $$p$$-power order and which preserves the $$G$$-conjugacy classes of elements of $$N$$ and of elements of prime power order is an inner automorphism (Theorem B). As a consequence, class-preserving automorphisms of a metabelian A-group are inner automorphisms, and if $$G$$ is a group which is an extension of an Abelian $$q$$-group by an Abelian $$q'$$-group for some prime $$q$$, then any automorphism of $$G$$ which preserves the conjugacy classes of elements of prime power order is an inner automorphism.
Finally, in Theorem C he proves that if $$G$$ is a finite group whose Sylow subgroups of odd order are cyclic, and whose Sylow $$2$$-subgroups are either cyclic, dihedral or generalised quaternion, then each class-preserving automorphism of $$G$$ is an inner automorphism.
These results are used to prove some variations of the Zassenhaus conjecture and to disprove a conjecture of Mazur by showing an A-group with a Coleman automorphism $$\sigma$$ of $$G$$ such that $$\sigma$$ is class-preserving and $$\sigma^2$$ is inner but $$\sigma$$ is not inner.

##### MSC:
 20D45 Automorphisms of abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
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