Studying p-groups via their Pfaffians: isomorphism testing and the PORC conjecture

Abstract: Given a field $K$, to each alternating $n \times n$ matrix of linear forms in $K[y_1,\dots ,y_d]$ one can associate a group scheme $\mathrm{G}$ over $K$. In particular, when $K$ is the field of rationals and $F$ is the field of $p$ elements, the $F$-points $\mathrm{G}(F)$ of $\mathrm{G}$ form a group of order $p^{n+d}$ and so, as $p$ varies, one obtains an infinite family of $p$-groups from $\mathrm{G}$. In this talk, I will present joint work with Josh Maglione and Christopher Voll, as well as ongoing work with Eamonn O'Brien, on the geometric study of automorphisms and isomorphism types of groups associated to small values of the parameters $n$ and $d$. I will also explain the implications of our work in connection to claims made around Higman's famous PORC conjecture.

group theory

Audience: researchers in the topic

Groups in Galway 2024