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Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permu …

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth …

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 $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ \mapsto …

A "near-miss" is $N(19328) = 19324$, while the only $n \leq 2000$ such that $|N(n)-n| \leq 25$ are $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $11$, $12$, $13$, $14$, $15$, $16$, $17$, $18$, $1 …

Let $n \in \mathbb{N}$. Is it true that for every $k \in \{1, \dots, n!\}$ there are some group $G$ and pairwise distinct elements $g_1, \dots, g_n \in G$ such that the set $\{g_{\sigma(1)} \cdot \ \d …

In order to answer the question we need a finite presentation of ${\rm SL}(3,\mathbb{Z})$ and a general method to find all subgroups of index $\leq n$ of a finitely presented group: A finite present …

Is it true that given a finitely presented group $G$, either all primes or only finitely many of them occur as orders of elements of $G$?

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series $\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely. Then there is a partition of the symmetric group ${\rm Sym}(\math …

This is also rather an expanded comment. -- Since for purely arithmetical reasons, $\ln(\ln(|G|))$ is a lower bound for the number $k(G)$ of conjugacy classes of a finite group $G$, maybe $$ f(G) := …

The answer to the question is yes, even if one additionally requires the group to be finitely generated. In this case, the question is Problem 9.10 in the Kourovka Notebook, and has been answered in: …

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permu …

Is it true that every finitely generated infinite simple group has exponential (word-)growth? Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of subexponent …

The conjugacy classes of elements of ${\rm SL}(2,\mathbb{Z})$ with given trace are counted in: S. Chowla, J. Cowles and M. Cowles: On the number of conjugacy classes in SL(2,Z). Journal of Number …

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, su …