Class-preserving automorphisms of finite groups.

*(English)*Zbl 0993.20017An 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.

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.

Reviewer: Adolfo Ballester-Bolinches (Burjasot)

##### 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 |

##### Keywords:

class-preserving automorphisms; inner automorphisms; A-groups; Frobenius groups; Sylow subgroups
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DOI

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