Complete group

In mathematics, a group, $G$ , is said to be complete if every automorphism of $G$ is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center.

Equivalently, a group is complete if the conjugation map, $G \to Aut(G)$ (sending an element $g$ to conjugation by $g$ ), is an isomorphism: injectivity implies the group is centerless, as no inner automorphisms are the identity, while surjectivity implies it has no outer automorphisms.

As an example, all the symmetric groups, $S n$ , are complete except when $n \in {2, 6}$ . For the case $n = 2$ , the group has a non-trivial center, while for the case $n = 6$ , there is an outer automorphism.

The automorphism group of a simple group, $G$ , is an almost simple group; for a non-abelian simple group, $G$ , the automorphism group of $G$ is complete.

A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the reverse need not hold: for example, the dihedral group of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see (Robinson 1996, section 13.5).

Assume that a group, $G$ , is a group extension given as a short exact sequence of groups

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