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 | ga = ag}. Prove that d divides |C(a)|.

Solution

By Lagrange;s theorem , the order d of any element a in a group H divides the order of H.

Let H = C(a).

Clearly a belongs to H, as a commutes with itself:: aa =aa.

So d = order of a divides order of C(a), as is to be proved.