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Let $X(G)=GC$ be a group, where $G$ is a semi dihedral group and $C$ is a cyclic group such that $G\cap C=1$. In this paper, $X(G)$ will be characterized.

In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer.

Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem…

Lindel\"of topological groups $G_1$ , $H_1$, $G_2$, $H_2$ are constructed in such a way that the products of $G_1 \times H_1$ and $G_2 \times H_2$ are not $\mathbb R$-factorizable groups and (1) the group $G_1 \times H_1$ is not…

Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition…

Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms $g, h\colon\Sigma^*\to\Delta^*$ there exists any non-trivial $x\in\Sigma^*$ such…

A result of Baumslag and Roseblade states that a finitely presented subgroup of the direct product of two free groups is virtually a direct product of free groups. In this paper we generalise this result to the class of cyclic subgroup…

Let $IA_n$ denote the group of $IA$-automorphisms of a free group of rank $n$, and let $\mathcal I_n^b$ denote the Torelli subgroup of the mapping class group of an orientable surface of genus $n$ with $b$ boundary components, $b=0,1$. In…

Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic…

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. In this paper we investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and…

In this paper we survey the main results about Golod-Shafarevich groups and their applications in algebra, number theory and topology.

Given a monomorphism $\Psi:\mathcal{H}\rightarrow \mathcal{F}$ where $\mathcal{H}$ is a proper free factor of the free group $\mathcal{F}$, we show the associated mapping torus $X$ of $\Psi$ has negative immersions iff $\mathcal{H}$ has…

Let $G$ be a $p$-group. We denote by $\mathcal{X}_i(G)$ the intersection of all subgroups of $G$ having index $p^i$, for $i \leq \log_p(|G|)$. In this paper, the newly introduced series $\{\mathcal{X}_i(G)\}_i$ is investigated and a number…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

Kn\"orr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his…

We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, decide whether the…

Let $G$ be a Polish group and let $H \leq G$ be a compact subgroup. We prove that there exists a Borel set $T \subset G$ which is simultaneously a complete set of coset representatives of left and right cosets, provided that a certain index…

A sequence $a_0<a_1<\ldots<a_n$ of nonnegative integers is called a Sidon sequence if the sums of pairs $a_i+a_j$ are all different. In this paper we construct CAT(0) groups and spaces from Sidon sequences. The arithmetic condition of Sidon…

In this note we study countable subgroups of the full group of a measure preserving equivalence relation. We provide various constraints on the group structure, the nature of the action, and on the measure of fixed point sets, that imply…

Random walks cannot, in general, be pushed forward by quasi-isometries. Tame Markov chains were introduced as a `quasi-isometry invariant' are a generalization of random walks. In this paper, we construct several examples of tame Markov…