11
$\begingroup$

Let $G$ be a finite group and $H_1,\ldots, H_n$ a set of maximal subgroups of $G$. Let $\delta_{H_i}$ be delta functions with support on $H_i$, and let $A$ be the commutative algebra generated by $\delta_{H_i}$.

Question: is dimension of $A$ greater or equal to $n$?

Motivation: In 1961 G.E.Wall conjectured that the number of maximal subgroups in a finite group $G$ is less than the order of $G$. A positive answer to the above question will imply Wall's conjecture. In fact it will even imply a relative version of Wall's conjecture: the number of maximal subgroups in $G$ which contain a subgroup $H$ is bounded by the number of double cosets in $G$, see arXiv:1006.5947 for more discussions and motivations from subfactors.

The question seems to be combinatorial in nature: the basis of $A$ is simply the list of subsets of $G$ obtained by cutting with $H_i$'s. for $n =2,3$ one can check directly, and one may try induction on $n$. Any references on this will be appreciated.

$\endgroup$

1 Answer 1

3
$\begingroup$

Wall's conjecture is now known to be false : see this question and the answers there.

$\endgroup$
1
  • $\begingroup$ But still open up to conjugacy classes thanks to the answer of Nick Gill. $\endgroup$ Commented Feb 19, 2014 at 13:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.