# Math Help - Prove C is a subgroup of G...

1. ## Prove C is a subgroup of G...

By the center of a group G we mean the set of all the elements of G which commute with every element of G, that is,

C = {a e G : ax = xa for every x e G}

Prove that C is a subgroup of G.

Any advice? I have no idea how to do this...

2. When proving a given set, $H$, is a subgroup there are two things you need to prove,

Given $a, b \in H$ then,

$a.b \in H$, and

$a^{-1} \in H$.

So, let $g \in G$ and $a, b \in Z(G)$, the center of $G$.

Does $g(ab) = (ab)g$? Yes, it does, but show it.

Does $ga^{-1} = a^{-1}g$? Yes, it does. However, there is a `trick' to this one. You need to remember that if $A=B$ then $A^{-1} = B^{-1}$, and that $ag^{-1} = g^{-1}a$ as $a \in Z(G)$.