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)What is the size of smallest Frobenius group $C_{q} \rtimes C_{p}$ satisfying the displayed equation?

2. What is the size of the smallest group satisfying the displayed equation?

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?

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 $q = \Theta(p^{2}/\log(p))$.

Questions:

1)What is the size of smallest Frobenius group $C_{q} \rtimes C_{p}$ satisfying the displayed equation?

2. What is the size of the smallest group satisfying the displayed equation?

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)What is the size of smallest Frobenius group $C_{q} \rtimes C_{p}$ satisfying the displayed equation?

2. What is the size of the smallest group satisfying the displayed equation?