Regular automorphism

An automorphism $ \phi $ of a group $ G $ such that $ g \phi \neq g $ for every non-identity element $ g $ of $ G $( that is, the image of every non-identity element of a group under a regular automorphism must be different from that element). If $ \phi $ is a regular automorphism of a finite group $ G $, then for every prime $ p $ dividing the order of $ G $, $ \phi $ leaves invariant (that is, maps to itself) a unique Sylow $ p $- subgroup $ S _ {p} $ of $ G $, and any $ p $- subgroup of $ G $ invariant under $ \phi $ is contained in $ S _ {p} $. A finite group that admits a regular automorphism of prime order is nilpotent (cf. Nilpotent group) [2]. However, there are solvable (cf. Solvable group) non-nilpotent groups admitting a regular automorphism of composite order.

#### References[edit]

[1] |  D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)

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[2] |  J.G. Thompson, "Finite groups with fixed-point-free automorphisms of prime order" Proc. Nat. Acad. Sci. , 45 (1959) pp. 578–581

#### Comments[edit]

A regular automorphism is also called a fixed-point-free automorphism.