We now consider three special types of subgroups.
Proposition
Let G be a group and X a subset of G. Then each of the following three subsets of G is a subgroup of G.
- The centralizer of X in G, i.e.,
the subset of all g
G with
gx = xg for all x of X.
- The normalizer of X in G, i.e., the subset of all
g G with
{gxg-1 | x X} = X.
- The center of G, i.e., the centralizer of G itself.
Note that the normalizer of a set X contains the centralizer as a subgroup. Both the normalizer and the centralizer contain the center of the group. Finally we note that the center is commutative.