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For various types of groups, there exist catalogues of those groups of the particular type which are "small" in a certain sense. — For example: The GAP Small Groups Library catalogizes groups of smal …

Given a finite group $G$, let $p(G)$ denote the number of prime factors of the order of $G$ (counting multiplicities). Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$ which grows fa …

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $c$ …

Let $G$ be a group which is generated by the set of its involutions, and assume that the product of every two involutions in $G$ has order a power of 2. Is it possible that $G$ has an element of odd o …

Given a finite group $G$, let $p(G)$ denote the largest prime factor of the order of $G$. For the purpose of this question, we say that the group $G$ has smooth order if its order exceeds the order of …

The order of a non-abelian finite simple group of order $n$ is bounded by $_2\log(n)$, and this bound is sharp in the sense that it cannot be improved by more than a small constant factor. See A bound …

A "randomly chosen" 2-generated dense subgroup $$ G \ = \ \langle a, b \rangle \ < \ {\rm A}_5 \times {\rm A}_6 \times {\rm A}_7 \times \dots $$ of the cartesian direct product of the finite simpl …

At least four such groups can share the same order: $$ |{\rm A}_7 \times {\rm PSp}(6,2)| = |{\rm PSU}(3,3) \times {\rm J}_2| = |{\rm A}_8 \times {\rm A}_9| = |{\rm PSL}(3,4) \times {\rm A}_9| …

Is it true that the order of a nonabelian finite simple group $G$ can be written as the product of the orders of two or more other nonabelian finite simple groups if and only if $G$ is either an alte …

One possible way of proving that a countable group $G$ is not finitely generated is finding an infinite set $S$ and a mapping $\varphi$ from $G$ to the power set of $S$ such that the following hold: …

Not necessarily. -- For example, the lamplighter group has exponential growth, but does not have a free subgroup of rank 2 (if $a$ and $b$ are two elements of that group of infinite order, then there …

Problem 20.30 in the Kourovka Notebook asks whether the maximum size of a conjugacy class of a perfect and centreless finite group $G$ is bounded below by $|G|^{\frac{1}{2}}$. Clearly, there cannot be …

The question has meanwhile been answered in the positive in: Joachim König, A note on the product of two permutations of prescribed orders. European Journal of Combinatorics 57 (2016), 50-56. The proo …

Today I noticed that the last relator in the 27-relator presentation of a group with unsolvable word problem given in Donald J. Collins: A simple presentation of a group with unsolvable word problem. …

The following is a question asked to me these days by Gülin Ercan. Let $G = L(q^f)$ be a finite simple group of Lie type, and let $L(q) \cong H \le G$ be the group of fixed points of the automorphisms …