### $\U0001d51bC$-elements in groups and Dietzmann classes.

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In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.

It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms ϕ and ψ is equal to the number of coincidence points of ϕ̂ and ψ̂ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally, we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.

Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n\left(G\right)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G$ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with...

We describe finite groups which contain just one conjugate class of self-normalizing subgroups.

We study infinite finitely generated groups having a finite set of conjugacy classes meeting all cyclic subgroups. The results concern growth and the ascending chain condition for such groups.

Making use of the Nielsen fixed point theory, we study a conjugacy invariant of braids, which we call the level index function. We present a simple algorithm for computing it for positive permutation cyclic braids.