6
votes
1answer
234 views

Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...
6
votes
2answers
498 views

Two groups acting on a set.

Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair ...
3
votes
1answer
387 views

Looking for deterministic criteria to generate the symmetric group?

So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its natural action on the set $T=\{1,2,\ldots,N\}$. Say that $H\leq S_N$ is a subgroup which acts ...
23
votes
2answers
1k views

Smallest n for which G embeds in $S_n$?

Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S_n$? Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations? This ...
4
votes
1answer
329 views

Centralisers of transitive permutation groups

Let $G$ be a transitive permutation group on a set of size $n$, and suppose $Z(G)=1$ (for instance $G$ is a direct power of a non-abelian simple group). What can we say about the centraliser $K$ of ...
31
votes
1answer
2k views

orders of products of permutations

Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...