Encyclopedia > Inner automorphism

  Article Content

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
East Islip, New York

... and 3,731 families residing in the town. The population density is 1,322.5/km² (3,428.5/mi²). There are 4,661 housing units at an average density of ...

 
 
 
This page was created in 34.6 ms