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What are the open problems concerning all the finite groups?

The references will be appreciated. Here are two examples:

  • Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy classes of maximal subgroups of a finite group is at most its class number (i.e. the number of conjugacy classes of elements, or the number of irreducible complex representations up to equiv.).

  • K.S. Brown's problem (B2000 Q.4; SW2016 p.760): Let $G$ be a finite group, $\mu$ be the Möbius function of its subgroup lattice $L(G)$. Then the sum $\sum_{H \in L(G)}\mu(H,G)|G:H|$ is nonzero.

There are two types of problems, those involving an upper/lower bound (like Aschbacher-Guralnick conjecture) and those "exact", involving no bound (like K.S. Brown's problem). I guess the first type is much more abundant than the second, so for the first type, please restrict to the main problems.

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    $\begingroup$ It's not clear what's a theorem/problem about "all" finite groups. A theorem about nilpotent finite groups, for instance, can provide information to Sylow subgroups of all finite groups, etc. Many problems about "all" finite groups are treated by reduction to special subcases (e.g., simple groups). $\endgroup$
    – YCor
    Mar 24, 2019 at 13:04
  • $\begingroup$ I would suggest not to restrict to "all" finite group - just finite groups would be ok, imho. It does not seems to me that we get so many answers that it would be difficult to navigate, and subdivision to "all" - "not all" would be justified $\endgroup$
    – Alexander Chervov
    Mar 24, 2019 at 17:09
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    $\begingroup$ I like this open problem (mathoverflow.net/q/316434/61536) on prime factrization of finite groups: Can each finite group $G$ be written as the product $G=A_1\cdots A_n$ of subsets of prime cardinality such that $|G|=|A_1|\cdots|A_n|$? $\endgroup$
    – Taras Banakh
    Mar 24, 2019 at 17:22
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    $\begingroup$ en.wikipedia.org/wiki/McKay_conjecture $\endgroup$
    – Dan Ramras
    Aug 28, 2020 at 5:28

1 Answer 1

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There are plenty of examples; the below are taken from the Kourovka Notebook, and include a fairly broad range of topics.

  • (4.55) Let $G$ be a finite group and $\mathbb{Z}_p$ the localization at $p$. Does the Krull-Schmidt theorem hold for projective $\mathbb{Z}_pG$-modules?
  • (8.51) If $G$ is a finite group and $p$ a prime, let $m_p(G)$ denote the number of ordinary irreducible characters of $G$ whose degree is not divisible by $p$. Let $P$ be a Sylow $p$-subgroup of $G$. Is it true that $m_p(G) = m_p(N_G(P))$?
  • (9.6) Is it true that an independent basis of quasiidentities of any finite group is finite?

A few other questions of a similar format to the questions you provided include (9.23), (11.17), (13.31), and (16.45).

The questions I picked are all of the form "Is $P$ true for all finite groups $G$?". There are many questions of the form "Characterize the finite groups $G$ for which $P$ holds", which to my ears has a slightly different ring to it than what you were asking for. There are also some questions regarding the class of finite groups, which again is a different interpretation.

Indeed, completely in line with Yves' comment, there is some ambiguity in what a problem for "all" finite groups should look like. Fortunately, there are enough problems in the Notebook to satisfy any interpretation, I should think.

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