Let G be group and a elementof G Show that a a1SolutionLet

Let G be group and a elementof G. Show that (a) = (a^-1).

Solution

Let G be a group, and let a G. .
If x C(a), ax = xa. Then x = a^-1ax = a-1xa and xa^-1 = a^-1xaa^-1 = a^-1x.
Thus x C(a^-1). Therefore C(a) C(a^-1). By applying the same idea for a^-1,
we have C(a^-1) C((a^-1)^-1) = C(a). Thus C(a) = C(a^-1).