Fixed-point subgroup

In algebra, the fixed-point subgroup of an automorphism f of a group G is the subgroup of G:[1]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle G^f = \{ g \in G \mid f(g) = 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 GS.

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.

To give an abstract example, let S be a subset 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.

See also

References

  1. ^ Checco, James; Darling, Rachel; Longfield, Stephen; Wisdom, Katherine (2010). "On the Fixed Points of Abelian Group Automorphisms". Rose-Hulman Undergraduate Mathematics Journal. 11 (2): 50.


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