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If $G$ is a group, then the center of $G$, assume its $Z(G)$, is the set of all elements of $G$ which commute with everything. So $Z(G)$ $=$ {$x$ $\in$ $G$: $xy$ $=$ $yx$ for all $y$ $\in$ $G$}. Prove that $Z(G)$ is a normal subgroup of $G$ for any group $G$.
First I need to show $G$ is a subgroup; and then show it's normal.
To show its a subgroup I know I need to prove that it has an identity, all elements contain inverses and it is closed under the operation
It should be clear that $1_G \in Z(G)$, but even if not, showing inverses and closure is enough to show $Z(G)$ contains the identity. For inverses: take arbitrary $x \in Z(G)$ and $g \in G$. Then we have:
$$x^{-1}g = (g^{-1}x)^{-1} = (xg^{-1})^{-1} = gx^{-1} \Rightarrow x^{-1} \in Z(G).$$
Can you do a similar thing to show that for $x,y \in Z(G)$, we have $xy \in Z(G)$? Showing it's normal is very easy once you use the commutativity property.
