# Complete group

2020 Mathematics Subject Classification: *Primary:* 20E *Secondary:* 22A [MSN][ZBL]

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 |

#### Comment

In topological group theory, a complete group may refer to a group that is a complete uniform space with respect to the uniformity implied by the topological group structure.

#### References

[a2] | Ross Geoghegan, Topological Methods in Group Theory, Graduate Texts in Mathematics 243, Springer (2008) ISBN 0-387-74611-0 |

**How to Cite This Entry:**

Complete group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Complete_group&oldid=35364