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Let $G$ be an infinite group. Let $N_0$ be the set of all $x\in G$ for which the conjugacy class $\{y^{-1}xy: y\in G\}$ is a finite set. Clearly $N_0$ is a normal subgroup. Iteratively, form an ascending transfinite sequence by

  • for $n$ a non-limit ordinal, let $N_n\subseteq G$ be the set of all $x\in G$ for which $xN_{n-1}\in G/N_{n-1}$ has finite conjugate class.
  • for $\omega$ a limit ordinal, $N_{\omega} = \bigcup_{\alpha<\omega} N_{\alpha}$.

There are two possibilities: either $(N_\alpha)$ stops at $G$, or at a proper subgroup $G^{FC}$ of $G$. For convenience, let's call (only for this post) $G^{FC}$ the [FC]-kernel of $G$, and $G$ a hyper-[FC] group if $G=G^{FC}$. A simple observation is that $G/G^{FC}$ is an [ICC] group. Let's call $G^{FC}$ a minimal [FC]-kernel if whenever $N_0\subseteq H\subseteq G$ is another normal subgroup for which $G/H$ is [ICC], we have $H\supseteq G^{FC}$.

I apologize if the questions below are elementary for MO.

Q0: Could you please point me to a reference where I can find the proper terminology for the made-up names hyper-[FC] group & [FC]-kernel?

Q1: Is $G^{FC}$ a hyper-[FC] group in itself?

Q2: Is $G^{FC}$ always minimal in the sense above?

Thanks in advance.


edit: Following Sean Eberhard's suggestion, I've just came across the book "Finiteness Conditions and Generalized Soluble Groups" by Robinson. The terminology he used in his book is FC-Hypercenter for $G^{FC}$, and FC-Hypercentral if $G=G^{FC}$. It is a big deal(!) that the made-up names by yours truly were a close call after all.. The book explains various central series in detail for those who are curious about the similarities in the construction & the properties of X-hypercenters, and more..

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  • $\begingroup$ Why is $N_0$ clearly a subgroup? $\endgroup$
    – LSpice
    Aug 8, 2023 at 13:54
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    $\begingroup$ @LSpice Let $O(x)=\{gxg^{-1}:g\in G\}$. For $x,y\in N_0$ and all $g\in G$, $gxy^{-1}g^{-1} = gxg^{-1}(gyg^{-1})^{-1}\in O(x)O(y)^{-1}$. So $O(xy^{-1})\subseteq O(x)O(y)^{-1} \Rightarrow |O(xy^{-1})|\leq |O(x)||O(y)|$. $\endgroup$
    – Onur Oktay
    Aug 8, 2023 at 14:15
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    $\begingroup$ $N_0$ is called the FC-center of $G$. groupprops.subwiki.org/wiki/FC-center $\endgroup$ Aug 8, 2023 at 14:32
  • $\begingroup$ @SeanEberhard thanks a bunch. I wonder if $(N_\alpha)$ is called [FC]-central sequence, analogous to central sequences. My made up name hyper-[FC] was inspired by the same anology. I'm asking for the right terminology so that I can use them as keywords for literature search. $\endgroup$
    – Onur Oktay
    Aug 8, 2023 at 15:04
  • $\begingroup$ @LSpice the FC-center $\mathrm{FC}(G)$ is a "clearly" subgroup because it consists of elements whose centralizer has finite index. $\endgroup$
    – YCor
    Aug 17, 2023 at 14:46

1 Answer 1

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Q0: For references try starting with Robinson's A course in the theory of groups, starting around 14.5.5. There the FC center is defined and some characterizations are given for FC groups (groups with finite conjugacy classes). From there it is not a great leap to define the second FC center and so on, but I am not aware of any significant study of these concepts.

The answers to Q1 and Q2 are both positive and use a simple observation. Suppose $N_1 \le N_2$ are normal subgroups of $G$ and consider the natural map $G/N_1 \to G/N_2$. Then the FC-center of $G/N_1$ maps into the FC-center of $G/N_2$. Indeed, if an element has a finite conjugacy class modulo $N_1$ then it certainly does so modulo the larger group $N_2$.

Q1: Let me write $F(G)$ for the FC-center and $F^\alpha(G)$ for the iterated variants. Let $N = \bigcup_\alpha F^\alpha(G)$ be the limit of the whole process. Then $N$ is a characterstic subgroup of $G$, as is $F^\alpha(N)$ for all $\alpha$. I claim inductively that $F^\alpha(G) \le F^\alpha(N)$ for all $\alpha$. The induction is obvious for zero and limit ordinals, and the case of successor ordinals follows from the observation above applied to $N/F^\alpha(G) \to N / F^\alpha(N)$.

Q2: Suppose $H$ is a normal subgroup of $G$ such that $G/H$ is ICC. I claim that $F^\alpha(G) \le H$ for all $\alpha$. Again it suffices to a consider a successor ordinal $\alpha+1$, and now we apply the observation above to $G / F^\alpha(G) \to G/H$, noting that $F(G/H)=1$.

In light of Q2, an alternative name for the "hyper FC center" might be the "ICC residual". It is the smallest normal subgroup $H$ such that $G/H$ is ICC.

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