Difference between revisions of "Complete group"
From Encyclopedia of Mathematics
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| − | A group | + | A group $G$ whose [[Centre of a group|centre]] is trivial (that is, $G$ is a so-called group without centre) and for which all automorphisms are inner (see [[Inner automorphism]]). The automorphism group of a complete group $G$ is isomorphic to $G$ itself (the term "complete" is related to this property). Examples of complete groups are the [[symmetric group]]s $S_n$ when $N \ne 2,6$. If a group $G$ contains a normal subgroup $B$ which is complete, then $G$ decomposes into a direct product $B \times K$ of the subgroup $B$ and its centralizer $K$ in $G$; indeed $K$ is isomorphic to the quotient group $G/B$. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, Yu.I. Merzlyakov, "Fundamentals of group theory" , Moscow (1982) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Hall jr., "Group theory" , Chelsea (1976)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov, Yu.I. Merzlyakov, "Fundamentals of group theory" , Moscow (1982) (In Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> M. Hall jr., "Group theory" , Chelsea (1976)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | <table> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> William Burnside, "Theory of Groups of Finite Order", 1911 ed. repr. Cambridge University Press (2012) ISBN 1108050328</TD></TR> | ||
| + | </table> | ||
Revision as of 18:24, 5 December 2014
A group $G$ whose centre is trivial (that is, $G$ is a so-called group without centre) and for which all automorphisms are inner (see Inner automorphism). The automorphism group of a complete group $G$ is isomorphic to $G$ itself (the term "complete" is related to this property). Examples of complete groups are the symmetric groups $S_n$ when $N \ne 2,6$. If a group $G$ contains a normal subgroup $B$ which is complete, then $G$ decomposes into a direct product $B \times K$ of the subgroup $B$ and its centralizer $K$ in $G$; indeed $K$ is isomorphic to the quotient group $G/B$.
References
| [1] | M.I. Kargapolov, Yu.I. Merzlyakov, "Fundamentals of group theory" , Moscow (1982) (In Russian) |
| [2] | M. Hall jr., "Group theory" , Chelsea (1976) |
| [a1] | William Burnside, "Theory of Groups of Finite Order", 1911 ed. repr. Cambridge University Press (2012) ISBN 1108050328 |
How to Cite This Entry:
Complete group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_group&oldid=35360
Complete group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_group&oldid=35360
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