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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.

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

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But still open up to conjugacy classes thanks to the answer of Nick Gill. –  Sébastien Palcoux Feb 19 at 13:09

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