What are the currently conjectures around symmetric group?

Question

I am asking you this question:

What are the currently conjectures around symmetric group on research?

Indeed I am interested to work on unsolved problems concerning symmetric or alternating groups.

I know it is a specific question and I asked it to a cultural point of view.

So don't hesitate if you have references of unsolved problems concerning symmetric or alternating groups.

Thanks.

Answer 1

There are some well-known conjectures concerning the symmetric group $S_n$. Here is one example, see Shalev:

Conjecture 1. The group $S_n$ has $n^{o(1)}$ conjugacy classes of primitive maximal subgroups.

Edit: This has been proved already, see below, but nevertheless is still an interesting topic.

Answer 2

OK, here is a conjecture that I want draw some attention to, not that I think it is easy.

Let $X(n)$ denote the multiset of irreducible character degrees of $S_n$. Let $m(n)$ denote the largest multiplicity amongst the elements of the set $X(n)$, i.e., the maximal multiplicity of a character degree of $S_n$.

Back in 2008, I proved that, $m(n)\geq n^{1/7}$ for all sufficiently large $n$.

Conjecture: There exists $\epsilon>0$ such that $m(n)<n^\epsilon$ for all sufficiently large $n$ (i.e., all $n$).