TITLE: Is there a reason why $Z(G)$ is named the "centre" of a group?
QUESTION [19 upvotes]: I just stumbled upon the definition of the center Z of a group G:
$$Z= \{x \in G \mid xz = zx \text{ for all } z \in G\}$$
The name “center” seems to suggest that there is some kind of geometric interpretation of the concept which I fail to see. My question is the following: is there some intuition/motivation behind the choice of naming $Z$ the “center” of a group?

REPLY [12 votes]: By virtue of left and right multiplications, a group $G$ "naturally lives" in $Sym(G)$ (the group of all the bijections of $G$ into itself) in the shape of a pair of subgroups of $Sym(G)$, say $\Theta$ and $\Gamma$, both of which it is isomorphic to. These subgroups commute, so $\Theta\Gamma$ is also a subgroup of $Sym(G)$. Finally, and this is mostly relevant for your question, $Z(G)$ turns out to be isomorphic to $\Theta \cap \Gamma$. Then, in $Sym(G)$ everything looks symmetric around the "center" $\Theta \cap \Gamma$:

For clarity, I'm not saying this is really the reason why the center was historically named that way. It's just a way I "pictorially" found for myself to accept that such a name actually makes sense.
Likewise, I've given here an interpretation of the wording "inner automorphism" (see Proposition 3 therein and the following comment).