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I just started reading about the upper and lower central series of a group. I'm wondering if there is a general relationship between the commutator subgroup $G'$ and the center $Z(G)$ of a group $G$.

There is Proposition that states that if $G'$ is not trivial and $G$ is nilpotent then $G'\cap Z(G)$ is not trivial. What if $G$ is not nilpotent?

More specifically, if the center of a group is trivial, can we deduce anything about the elements of $G'$?

Teplotaxl
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    No, even if $Z(G)=1$, then $[G,G]$ still can be quite different. Take $G=S_3$ and $G=A_5$, for example. – Dietrich Burde Jan 10 '22 at 20:07
  • Even in special cases it is difficult to come up with a general relationship. See for example this recent post. Of course there are some relationships, e.g., the one given in this post: "In any group $G$, we have $[G,G]\cap Z(G)\subseteq \operatorname{Frat}(G)$". So your question needs more focus. – Dietrich Burde Jan 10 '22 at 20:10
  • In some sense the commuter is a measure of how far two elements are from being commutative. If the center is trivial you should expect rich behavior from the commuter subgroup. – CyclotomicField Jan 10 '22 at 20:11
  • For infinite finitely generated groups: if the center is not of finite index, then the commutator subgroup is infinite. – markvs Jan 10 '22 at 20:14
  • I'm sorry, I meant to write what if $G$ is not nilpotent. – Teplotaxl Jan 10 '22 at 20:24
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    If $G$ is not nilpotent then the claim $G'\cap Z(G)\neq 1$ is wrong. Again, take $G=S_3$. This group is solvable but not nilpotent. – Dietrich Burde Jan 10 '22 at 20:28
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    At a high level one can say that $G'$ and $G/Z(G)$ are a kind of "dual". It would take me to far to explain why this is the case and what that exactly means. The theory of isoclinisms (https://en.wikipedia.org/wiki/Isoclinism_of_groups) of groups is a start. One nice fact, due to Isaac Schur: if $G/Z(G)$ is finite then $G'$ is finite. And the reverse holds if for example $G$ is finitely generated. – Nicky Hekster Jan 11 '22 at 10:25
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    On @NickyHekster's "duality" comment, you can see this question/answer. – Arturo Magidin Feb 16 '22 at 19:44
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    What a great answer @ArturoMagidin! This might also be interesting: https://bit.ly/3HXc3bl (old paper of mine about Varieties of Groups and Isologisms, where a lot of this "duality" is present w.r.t. varieties of groups). – Nicky Hekster Feb 17 '22 at 11:11
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    @NickyHekster Thanks! I was not aware of that paper, which is a bit surprising since at one point I did an extensive search for papers about varieties, and this was published less than 10 years before I did that. Then again, at the time I was using the CD-ROM version of MathReviews/MathSciNet, so that may be how I missed it. – Arturo Magidin Feb 17 '22 at 15:15
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    Those were the good old days spending hours in the library browsing journals … CD-ROM was pretty advanced! – Nicky Hekster Feb 17 '22 at 15:28

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