Group Automorphism


A group automorphism is an isomorphism from a group to itself. If G is a finite multiplicative group, an automorphism of G can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity G={1,-1,i,-i} can be written as shown above, which means that the map defined by

 1|->1,    -1|->-1,    i|->-i,    -i|->i

is an automorphism of G.

The map f(x)=nx is also a group automorphism for Z/pZ as long as n is not congruent to 0. Conjugating by a fixed element h is a group automorphism called an inner automorphism.

In general, the automorphism group of an algebraic object O, like a ring or field, is the set of isomorphisms of that object O, and is denoted Aut(O). It forms a group by composition of maps. For a fixed group G, the collection of group automorphisms is the automorphism group Aut(G).

See also

Automorphism, Automorphism Group, Finite Group, Group, Inner Automorphism, Isomorphism, Multiplication Table Outer Automorphism

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by Todd Rowland

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Cite this as:

Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "Group Automorphism." From MathWorld--A Wolfram Web Resource.

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