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]](https://en.wikipedia.org/wiki/Infinite_conjugacy_class_property) 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.




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**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..



Finite conjugacy classes
**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.


