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
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 AutomorphismPortions 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 Resource. https://mathworld.wolfram.com/GroupAutomorphism.html