3
$\begingroup$

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 \begin{equation}\label{Syl} (P_{1}\cdot P_{2}) \cap P_{3} = \{1_{G}\} \,. \end{equation}

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Nick Gill Dec 19 '17 at 12:14
  • $\begingroup$ 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 $\endgroup$ – Padraig Ó Catháin Dec 22 '17 at 12:37
  • $\begingroup$ @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 $|H|\geqslant |G|^{1/3}$. $\endgroup$ – Glasby Feb 9 '18 at 13:39
4
$\begingroup$

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 $P_1P_2\cap P_3=\{1,a,ab,b\}\cap\{1,ab^2\}=\{1\}$.

$\endgroup$
  • $\begingroup$ 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 $p$. $\endgroup$ – Padraig Ó Catháin Feb 9 '18 at 13:05

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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