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The sharply $k$-transitive groups of finite degree $n$ ($k$-transitive of order $n(n-1)\cdots (n-k+1)$) were investigated by Zassenhaus in the 1930s. You can find a modern account for example in Chapt …

It is not hard in practice, at least for reasonably small values of $m,n,k$, to find permutations with the required property, and I did once convince myself that I could find a systematic way of doing …

Let me have a go at this one. We want to show that for any finite set $W$ of reduced words in $a,b$, we can find a subgroup $G = \langle a,b \rangle$ of some symmetric group $S_k$ such that all words …

According to my calculations, the minimal permutation degrees of $2.A_n$ for $n=5,\ldots, 24$ are as follows. In each case $H$ is a core-free subgroup of $2.A_n$ of largest order. The minimal permutat …

Concerning the second part of the OP's question: Also, how can you efficiently find the size of the subgroup $\langle a,b \rangle$ in $S_n$ ? My crude tests consists of randomly multiplying the …

The following seems to be a counterexample, verified by a Magma calcualtion. The vertex set is $\{1,2,\ldots,16\}$ and the edge set is $E$. The quotient graph is the complete graph on four vertices an …

I think we can get all transpositions with words at length at most 5 over $\Sigma_n$, which will give an upper bound of $5(2^n-1)$. This can easily be improved slightly - for example, you can use elem …

It is proved in Holt, D. F., Enumerating perfect groups. J. London Math. Soc. (2) 39 (1989), no. 1, 67–78 that $n^{2l(n)^2/27−dl(n)} \le F(n) \le n^{l(n)^2/6+l(n)}$ for some constant $d$, where $l …

I generally agree with Geoff that the question is too broad, and that you should tell us what types of families of subgroups you are most interested in. I don't know of any proven results in this dir …

I can do a bit better! For even $n=2m$ there is a subgroup of order $2^{m-1}m!$ with no 2-cycles or 3-cycles. Let $W$ be the wreath product of a cyclic group of order 2 with $S_m$. In other words, $W …

There is a large number of distinct (up to module isomorphism) ${\mathbb F}_pG$-modules, even in small dimensions, and I don't think you can hope for any kind of reasonable classification. I did some …

I am just turning my comment into an answer. The answer to the question is no. I think the highest possible value of $r_2(g)$ with $g \ne 1$ is $r_2(g) = 3/4$ for the central element $g$ of $Q_8$, but …

This is a very crude attempt to explain the growth rate of $13$, can be almost certainly be improved upon. In the situation where $20$ is the diameter, there are $18$ moves (or monoid generators), w …

I am afraid that I would have no idea where to look for a reference for this statement, but here is a very rough sketch proof. I can fill in details if necessary. Let $A_1,\ldots,A_s$ be the orbits o …

As has been pointed out in comments, the only subgroups $G$ and $H$ of $S_{10}$ isomorphic to $S_7$ and $S_6$ that could possibly have trivial intersections are the copy of $S_7$ that lies in $A_{10}$ …