TITLE: subgroups of the centers of a group are all normal
QUESTION [5 upvotes]: Show a proof that $Z(G),$ the center of $G,$ is a normal subgroup of $G$ and every subgroup of the center is normal. I know that $Z(G)$ is a normal subgroup and it is abelian, but how would I show that every subgroup of $Z(G)$ is also normal?

REPLY [4 votes]: Recall that the center of a group $Z(G)$ consists of all elements that commute with every element in $G$. That's pretty much all you need, here:
If $H<Z(G),$ and $h \in H,$ then $h \in Z(G),$ and so for every $g \in G,\;$ $g^{-1}hg =g^{-1}gh = eh = h$ ($h \in Z(B)$, so $h$ commutes with $g$.)
Since this is true of all $h \in H$, we have that $g^{-1}Hg = H$, and hence, $H$ is normal.