Difference between revisions of "Regular automorphism"
From Encyclopedia of Mathematics
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| + | An [[Automorphism|automorphism]] $ \phi $ | ||
| + | of a [[Group|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|Nilpotent group]]) [[#References|[2]]]. However, there are solvable (cf. [[Solvable group|Solvable group]]) non-nilpotent groups admitting a regular automorphism of composite order. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.G. Thompson, "Finite groups with fixed-point-free automorphisms of prime order" ''Proc. Nat. Acad. Sci.'' , '''45''' (1959) pp. 578–581</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.G. Thompson, "Finite groups with fixed-point-free automorphisms of prime order" ''Proc. Nat. Acad. Sci.'' , '''45''' (1959) pp. 578–581</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
A regular automorphism is also called a fixed-point-free automorphism. | A regular automorphism is also called a fixed-point-free automorphism. | ||
Latest revision as of 08:10, 6 June 2020
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
| [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.
How to Cite This Entry:
Regular automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_automorphism&oldid=17880
Regular automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_automorphism&oldid=17880
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article