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In GAP, you can find the number of elements of order $t$ in ${\rm GL}(2,q)$ by the following function: NumberOfElementsOfGivenOrderInGL2q := function ( q, t ) return Sum(List(Filtered(ConjugacyClas …

As Geoff Robinson has already said, the answer to the question is no. In dimension $4$, there are in total $120$ counterexamples, of which $96$ have kernel of dimension $1$, and $24$ have kernel of di …

If I understand your question right, your group $G$ has order $5160960$, and it has an elementary abelian normal subgroup $N$ of order $2^7$ such that $G/N \cong {\rm S}_8$. This can be found with GA …

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 …

Let $f$ denote the function described in the question. The assertion that every trajectory of $f$ except for the one starting at 0 ends in the cycle -1, 1, -1 is equivalent to the Collatz conjecture s …

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 …

There are some quantifiers unclear in your question, but regardless of how to read it, your assertion is false. -- The smallest counterexample with blocks of pairwise distinct size all of whose eigen …

GAP provides a function NullspaceIntMat which solves systems of linear diophantine equations. The documentation says: 25.1-2 SolutionIntMat * SolutionIntMat( mat, vec ) ───────────────────────────── …

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 …