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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.

Lazare
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    I guess mathoverflow might be a more suitable place to ask this question. – Brozovic Jul 31 '20 at 17:59
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    I'm not sure MO is a good place for the question either. 'What are open problems in finite groups?' is a pretty wide-open question. – David A. Craven Jul 31 '20 at 18:05
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    I think it would be quickly closed on MO as being far too broad. – Derek Holt Jul 31 '20 at 18:22
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    I don’t know if you’ve looked at the Kourovka Notebook, which is regularly updated. The last notice of update I have is from June 24th. – Arturo Magidin Jul 31 '20 at 19:42
  • Thanks for the link to kourovka notebook, there isn't a lot of about symmetric and alternating group. But thanks it is interessing! – Lazare Jul 31 '20 at 22:03
  • I have edited the original post of this topic to reopened it more precisely on symmetric and alternating groups. May be somebody could reopened it? – Lazare Jul 31 '20 at 22:04
  • @Lazare I do think this still needs more focus. I am not convinced that this is the place for problem lists, as lots exist already. So you need a new "twist" to the question. For example, you could ask what problem lists there are, or you could ask about problems which are particularly influential. Or something. – user1729 Aug 01 '20 at 09:33

2 Answers2

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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 $n$).

David A. Craven
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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.

Dietrich Burde
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    This was proved by Liebeck-Martin-Shalev in 2005. – David A. Craven Jul 31 '20 at 18:20
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    To add to my short comment, in their paper, L-M-S proved that the number of maximal subgroups in 'curly S', i.e, ones not coming from positive-dimensional subgroup, are bounded in terms of the Lie rank only. This is enough to prove that result. – David A. Craven Jul 31 '20 at 18:24