1.20: Centralizer
The centralizer C G (g) of an element g of a group G is the set of elements of G which commute with g:
- C G (g) = {x ∈ G : xg = gx}.
If H is a subgroup of G, then C H (g) = C G (g) ∩ H.
More generally, if S is any subset of G (not necessarily a subgroup), the centralizer of S in G is defined as
- C G (S) = {x ∈ G : ∀ s ∈ S, xs = sx}.
If S = {g}, then C(S) = C(g).
C(S) is a subgroup of G; in fact, if x, y are in C(S), then xy −1 s = xsy −1 = sxy −1 .