Skip to main content

All Questions

Filter by
Sorted by
Tagged with
7 votes
1 answer
344 views

For which $n$ can $S_n$ act transitively on $n+k$ elements?

It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$. Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...
M. Winter's user avatar
  • 13.6k
31 votes
2 answers
1k views

Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
John McVey's user avatar
  • 1,068
16 votes
2 answers
722 views

Minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
Igor Rivin's user avatar
  • 96.4k
2 votes
0 answers
156 views

Special sets of involutions generating ${\rm S}_n$

For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$ $(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?: for any $n$, the $g_{n,i}$ $(i = 1, \dots, ...
Stefan Kohl's user avatar
  • 19.6k
5 votes
1 answer
441 views

Minimum word length for an unusual set of generators of the symmetric group

Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form $$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$ Find the least integer $f_n$...
Bhaskar Vashishth's user avatar