Let a be an element of order d in a finite group G Recall th

Let a be an element of order d in a finite group G. Recall that the centralizer of a in G is the subgroup C (a) = {g elementof G l ga = ag}. Prove that d divides IC(a)|.

Solution

Recall (following Lagrange\'s theorem) that the order d of an element a in a finite group H divides the order of H.

Let H = C(a).

Now C(a) is a group and a belongs to C(a) as a commutes with a. a.a =a.a

Applyig the above , we get d divides order of H = order of C(a)