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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 $c$ …

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 $G$?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which …

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 …

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite? If the answer is yes, can $P …

There are indeed other such polygons. -- For example there is one for $n = 11$, as follows (the origin is in the lower left corner): Also there is one for $n = 15$: Further there are $21$ such p …

Definition / Question Definition: 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 $\t …

Let $G \leq {\rm S}_n$ be a finite permutation group, and let $S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed under inversion and which does not contain the identity. The growth …

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. …

For small enough $n$, an efficient way to perform this enumeration is described in the solution to a GAP exercise I posed a few years ago. It basically amounts to setting up a suitable matrix, raising …

Let $n$ be a positive integer, and let $n = \ell_1 + \dots + \ell_k$ be a partition of $n$. Then there exists a convex polygon with side lengths $\ell_1, \dots, \ell_k$ if and only if all of the $\ell …

If you are also interested in problems of that type where $n = \infty$: Given a mapping $f: \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to themselves, it is often a notoriously hard pr …

Yes, there is a common name for such graphs -- they are called graphs with polynomial growth. See e.g. W. Imrich, N. Seifter: A survey on graphs with polynomial growth, Discr. Math. 95 (1991), 101-11 …

On page 218 of John D. Dixon, Brian Mortimer: Permutation Groups, Springer GTM 163, 1996 it is stated: It is a consequence of the classification of finite simple groups that a finite permutat …