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?
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4$\begingroup$ You may probably start by asking if there is a reason that a group is named a "group." :) $\endgroup$– AlexeyJan 15, 2019 at 23:50
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1$\begingroup$ Related-hsm.stackexchange.com/questions/7686/… $\endgroup$– cqfdJan 25, 2019 at 9:54
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1$\begingroup$ @ThomasShelby Thanks! But its a similar situation: after some digging, nothing concrete was found. $\endgroup$– user1729Jan 25, 2019 at 11:11
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$\begingroup$ "Central" also means "of primary importance", so maybe central elements of a group where introduced as "central" because they were of primary importance in the text that introduced them. $\endgroup$– LazarusJan 25, 2019 at 11:32
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3 Answers
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An element is called central if it commutes with everything else...i.e., it does not matter whether you multiply from the left or right, so you can think of such an element as being multiplied in the "center" of any product it is in. Starting from there, it is an easy step to start calling the subgroup of all such elements the center. And from there we call it $Z(G)$, the Z being an abbreviation for the German word for center if I remember right.
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2$\begingroup$ Okay but why are those elements called central? $\endgroup$– D_SJan 15, 2019 at 22:44
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4$\begingroup$ @D_S Because when left- and right-multiplication agree, you can think of the operation as a third and honorary "central" kind of multiplication, as if $x$ were written literally on top of $z$. $\endgroup$– J.G.Jan 15, 2019 at 22:51
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3$\begingroup$ Yep, it stands for "Zentrum" in German, which just means center. $\endgroup$– zxmknJan 15, 2019 at 23:15
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8$\begingroup$ Is this just an educated guess, or do you have a (historical?) citation? $\endgroup$– user1729Jan 16, 2019 at 11:34
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1$\begingroup$ I would suggest that if one want to go back further than the references given in the HSM article, one would want to look at the literature on Lie algebras. It's fairly transparent that the term "nilpotent" was first used for Lie algebras and then applied to groups. So what about the term "lower central series"? If that was also imported from Lie algebras, then Lie algebras might have been the source for de Séguier's terminology. Not sure this will pan out, but I don't have access to pre-1905 literature of Lie algebras, so I can't check it myself. $\endgroup$ Jan 25, 2019 at 12:55
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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).
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1$\begingroup$ Lovely! Though I guess that $\Gamma^{\operatorname{op}}$, rather than $\Gamma$, is isomorphic to $G$. $\endgroup$– user1007416Feb 23, 2022 at 20:05
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Since
$$xz = zx \iff x = zxz^{-1}$$
$Z$ can also be written as
$$ Z = \{x \in G \mid x = zxz^{-1} \text{ for all } z \in G \}$$
I hope the name is more intuitive now!
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1$\begingroup$ Out of the three answers so far, I am buying this one, i.e. $x$ is always between an element and the inverse of that element. $\endgroup$– NobodyJan 16, 2019 at 8:35