Regular automorphism

An automorphism of a group such that for every non-identity element of (that is, the image of every non-identity element of a group under a regular automorphism must be different from that element). If is a regular automorphism of a finite group , then for every prime dividing the order of , leaves invariant (that is, maps to itself) a unique Sylow -subgroup of , and any -subgroup of invariant under is contained in . 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

[1]	D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)
[2]	J.G. Thompson, "Finite groups with fixed-point-free automorphisms of prime order" Proc. Nat. Acad. Sci. , 45 (1959) pp. 578–581

Comments

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