a) Suppose . We want to show i.e. .
Take .
b) Suppose . We need , so we want to prove that we have .
Take .
c)" " Suppose . We want i.e. . Take .
" ": From b) we know .
Let G be a group. Let the centralixer C(X), X a subset of G, be defined as follows: C(X)={g in G | gx=xg for all x in X}.
Prove the following:
If X is a subset of Y, then C(Y) is a subset of C(X)
X is a subset of C(C(X))
C(X) = C(C(C(X)))