Narrow Results

For any group $G$, let $\mathrm{\text{cent}}(G)$ denote the set of all centralizers of $G$. K. Khoramshahi and M. Zarrin [Groups with the same number of centralizers, J. Algebra Appl. 20 (2021) 2150012], posed the following conjecture: Let $G$ and $S$ be finite groups. Is it true that if $|\mathrm{\text{Cent}}(G)|=|\mathrm{\text{Cent}}(S)|$ and $\left|G^{\prime }\right|=\left|S^{\prime }\right|$, then $G$ is isoclonic to $S$? In this paper, among other things, we give a negative answer to this conjecture.

The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer $r$ we prove that if $G$ is a profinite group in which the centralizer of every nontrivial element has rank at most $r$, then $G$ is either a pro-$p$ group or a group of finite rank. Further, if $G$ is not virtually a pro-$p$ group, then $G$ is virtually of rank at most $r+1$.

For a group $G$, we say that $x,y\in G$ are in the same $z$-class if their centralizers in $G$ are conjugate. The notion of $z$-class has origin in a connection between geometry and groups. However, as the notion is purely group theoretic, in this paper, we focus our attention on the influence of the $z$-classes on the structure of the group. The number of $z$-classes is invariant for a family of isoclinic groups. We obtain bounds for the number of $z$-classes in certain families of groups. A non-abelian finite $p$-group contains at least $p+2$$z$-classes. Moreover, we characterize the non-abelian $p$-groups with $p+2$$z$-classes; these are precisely, up to isoclinism, the $p$-groups of maximal class with an abelian subgroup of index $p$.

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type $F_{\infty }$) counterexample based on Thompson’s group $F$. We construct as well a finitely generated counterexample with finite decomposition complexity.