Regular automorphism
An automorphism
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
[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.
Regular automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_automorphism&oldid=48478