View source for Complete 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====
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<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>
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<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>
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