In
algebra, the fixed-point subgroup
of an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
''f'' of a
group ''G'' is the
subgroup of ''G'':
:
More generally, if ''S'' is a
set of automorphisms of ''G'' (i.e., a subset of the
automorphism group of ''G''), then the set of the elements of ''G'' that are left fixed by every automorphism in ''S'' is a subgroup of ''G'', denoted by ''G''
''S''.
For example, take ''G'' to be the group of
invertible ''n''-by-''n''
real matrices and
(called the
Cartan involution). Then
is the group
of ''n''-by-''n''
orthogonal matrices
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
.
To give an abstract example, let ''S'' be a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of a group ''G''. Then each element ''s'' of ''S'' can be associated with the automorphism
, i.e.
conjugation by ''s''. Then
:
;
that is, the
centralizer of ''S''.
References
{{algebra-stub
Algebraic groups