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The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: Ab …

Let $a,b$ be two positive integer numbers. A group $G$ is called $a{\times}b$-decomposable if there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$ where $AB=\{xy:x\in A …

Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subse …

The answer here is negative. In fact, any non-trivial quotient group of the symmetric group $\mathrm{Sym}(X)$ contains a copy of $\mathrm{Sym}(X)$. Indeed, by the Baer-Schreier-Ulam Theorem, any norm …

Question. Is it true that each uncountable group $G$ contains an uncountable subgroup $A$ and an infinite subgroup $B$ such that $A\cap B=\{1\}$? What will be the answer if we additionally require tha …

Let us recall that a group $G$ is called perfect if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial. Question. Is there any name for a grou …

This question is a development of my previous question. Let $G$ be a finitely generated group acting transitively on an infinite set $X$ so that for every $g\in G$ and $x\in X$ the $g$-orbit $\{g^nx: …

Looking at the von Neumann–Bernays–Gödel (NBG) axioms of Set Theory in Wikipedia I noticed that it has two axioms of permutation: one for circular permutations and another for transpositions. On the …

Since I was not satisfied with the answers and comments obtained sofar, I decided to think on this question myself. Here is the list of class existence axioms of NBG (written in an informal form): …

Maybe too late as for a question asked 11 years ago, but I would like to inform that this paper contains a topological proof of the following theorem which is related to the answer of Mariano Suárez-Á …

A group $G$ is called $\bullet$ topologizable if $G$ is algebraically isomorphic to a non-discrete Hausdorff topological group; $\bullet$ nontopologizable if $G$ is not topologizable; $\bullet$ bounde …

I need to consider finite groups $G$ such that for any square-free number $d$ dividing the order of the group $G$, there exists a normal subgroup $H$ in $G$ such that either $H$ or $G/H$ has order $d …

It is known that the number of ends of a finitely generated group is 0,1, 2 or $\infty$. Problem 1. What is known about the number of ends of infinite finitely generated torsion groups? In parti …

In his famous paper "On a problem of Kurosh, Jonsson groups, and applications" of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6640}$ for any uncountable subset $A\su …

In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $ …