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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