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Automorphism orbits of finite groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Thomas J. Laffey  and
Desmond MacHale
Thomas J. Laffey
Affiliation:
Department of Mathematics, University College, Dublin, Ireland
Desmond MacHale
Affiliation:
Department of Mathematics, University College, Cork, Ireland
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Abstract

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Let G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

MSC classification

Secondary: 20F28: Automorphism groups of groups 20E36: Automorphisms of infinite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 2 , April 1986 , pp. 253 - 260
Copyright
Copyright Australian Mathematical Society 1986

References

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