# Intersections of products of Sylow $p$-subgroups

Motivated by the work of Cohn, Umans and collaborators on bounding the complexity of matrix multiplication, a student and I have been investigating the following problem. For subsets $X$ and $Y$ of a group $G$, let $X \cdot Y = \{ xy \mid x \in X, y \in Y \}$.

We would like to find the size of the smallest group with three distinct Sylow $p$-subgroups $P_{1}, P_{2}$ and $P_{3}$ such that $$\label{Syl} (P_{1}\cdot P_{2}) \cap P_{3} = \{1_{G}\} \,.$$

Any group having this property has at least $p+1$ Sylow $p$-subgroups, and so has order greater than $p^{2}$, while $\mathrm{PSL}_{2}(p)$ is an explicit example of a group with the required property. So $p^{2} < |G| < p^{3}$, we are interested in whether there exists a stronger bound of the form $|G|< p^{3-\epsilon}$ for some absolute $\epsilon > 0$ (not tending to $0$ with $p$).

We tried restricting to Frobenius groups $G = C_{q} \rtimes C_{p}$ where $q = kp+ 1$ is prime. Computational evidence suggests that for $p > 5$ such groups exist with $q < p^{2}$. Realising $G$ as a group of linear polynomials with coefficients in $\mathbb{F}_{q}$ we can reformulate the problem of estimating the intersection of point stabilisers $(G_{0} \cdot G_{1}) \cap G_{t}$ as counting the number of solutions of the polynomial $x^{k} + (t-1)y^{k} - t = 0$, for some $t \in \mathbb{F}_{q}$. The Hasse-Weil bound gives a lower bound for $q$ in terms of $p$, but these are weaker than we would like. If we assume that the fixed points of elements in $G_{0} \cdot G_{1}$ are distributed close to uniformly at random in $G$, then we have a heuristic argument that a Frobenius group satisfies the displayed equation only when $q = \Omega(p^{2}/\log(p))$.

Questions:

1)For a given prime $p$, what is the size of smallest Frobenius group $C_{q} \rtimes C_{p}$ satisfying the displayed equation?

2) For a given prime $p$, what is the size of the smallest group satisfying the displayed equation?

• I seem to recall that there is a theorem that characterizes the simple groups $G$ which have a Sylow $p$-subgroup with order at least $\sqrt[3]{|G|}$. (This article comes close -- iopscience.iop.org/article/10.1070/SM1982v043n03ABEH002453 -- but I can't access it to check it gives the full details.) Anyway, I believe that the only such FSG's are $PSL_2(q)$ and $Suz(q)$ -- presumably this could be checked reasonably easily using CFSG. My feeling is that this could be used to show that any family of groups $G$ which (asymptotically) satisfies $|G|<p^{3-\varepsilon}$ must be solvable. – Nick Gill Dec 19 '17 at 12:14
• Thanks Nick - this is helpful! I just saw a paper on the arxiv which deals with cross characteristic cases: arxiv.org/pdf/1712.05899.pdf – Padraig Ó Catháin Dec 22 '17 at 12:37
• @Nick the paper Alavi, Seyed Hassan; Burness, Timothy C. Large subgroups of simple groups. J. Algebra 421 (2015), 187–233 characterizes pairs $(G,H)$ with $G$ simple and $|H|\geqslant |G|^{1/3}$. – Glasby Feb 9 '18 at 13:39

The answer to your both questions is 6. Consider the symmetric group $S_3=\langle a,b\mid a^2=b^3=1, b^a=b^{-1}\rangle$, and take $P_1=\langle a\rangle$, $P_2=\langle ab\rangle$, $P_3=\langle ab^2\rangle$. Then $P_1P_2\cap P_3=\{1,a,ab,b\}\cap\{1,ab^2\}=\{1\}$.
• Thanks for this Steve! We can compute the solutions to both problems for small groups, but we haven't been able to show that for large $p$ there is a solution of size $O(p^{3-\epsilon})$ for $\epsilon >0$. I've clarified the question to ask for all primes $p$. – Padraig Ó Catháin Feb 9 '18 at 13:05