Encyclopedia > Inner automorphism

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Inner automorphism

In abstract algebra, if G is a group and a is an element of G, then the function f : G -> G defined by
f(x) = axa-1     for all x in G
is called an inner automorphism of G. As the name suggests, f is a group automorphism of G.

The collection of all inner automorphisms of G forms a normal subgroup of the full automorphism group G. This group is denoted by Inn(G).

By associating the element a in G with the inner automorphism f in Inn(G) as above, one obtains an isomorphism between the factor group G/Z(G) (where Z(G) is the center of G) and Inn(G). As a consequence, the group of inner automorphisms Inn(G) is trivial (i.e. consists only of the identity element) if and only if G is abelian.

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