Complete group

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A group whose centre (cf. Centre of a group) is trivial (that is, is a so-called group without centre) and for which all automorphisms are inner (see Inner automorphism). The automorphism group of a complete group is isomorphic to itself (the term "complete" is related to this property). Examples of complete groups are the symmetric groups when (cf. Symmetric group). If a group contains a normal subgroup which is complete, then decomposes into a direct product of a subgroup and its centralizer in .


[1] M.I. Kargapolov, Yu.I. Merzlyakov, "Fundamentals of group theory" , Moscow (1982) (In Russian)
[2] M. Hall jr., "Group theory" , Chelsea (1976)
How to Cite This Entry:
Complete group. Encyclopedia of Mathematics. URL:
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