Fixed-point subgroup

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the fixed-point subgroup 
 
 
 
 
 G
 
 f
 
 
 
 
 {\displaystyle G^{f}}
 
 of an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> f of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G is the <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of G:

G
          
            f
          
        
        =
        {
        g
        ∈
        G
        ∣
        f
        (
        g
        )
        =
        g
        }
        .
      
    
    {\displaystyle G^{f}=\{g\in G\mid f(g)=g\}.}

More generally, if S is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of automorphisms of G (i.e., a subset of the <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> 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 <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> n-by-n <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> and 
 
 
 
 f
 (
 g
 )
 =
 (
 
 g
 
 T
 
 
 
 )
 
 −
 1
 
 
 
 
 {\displaystyle f(g)=(g^{T})^{-1}}
 
 (called the <a href="/facts/Cartan_involution/G8g40WGg">Cartan involution</a>). Then 
 
 
 
 
 G
 
 f
 
 
 
 
 {\displaystyle G^{f}}
 
 is the group 
 
 
 
 O
 (
 n
 )
 
 
 {\displaystyle O(n)}
 
 of n-by-n <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrices</a>.
To give an abstract example, let S be a <a href="/facts/Subset/SJGERmbA">subset</a> of a group G. Then each element s of S can be associated with the automorphism 
 
 
 
 g
 ↦
 s
 g
 
 s
 
 −
 1
 
 
 
 
 {\displaystyle g\mapsto sgs^{-1}}
 
, i.e. <a href="/facts/Conjugacy_class/AFuPIFNW">conjugation</a> by s. Then

G
 
 S
 
 
 =
 {
 g
 ∈
 G
 ∣
 s
 g
 
 s
 
 −
 1
 
 
 =
 g
 
  for all 
 
 s
 ∈
 S
 }
 
 
 {\displaystyle G^{S}=\{g\in G\mid sgs^{-1}=g{\text{ for all }}s\in S\}}
 
;
that is, the <a href="/facts/Centralizer/AKhrEUGR">centralizer</a> of S.

Fixed-point subgroup open-in-new

Fixed-point subgroup