Extension of elementary $p$-groups and its application in classification of groups of prime exponent

Abstract

Let $p$ be a prime number and ${\mathbb{Z}}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence between the following two sets: (i) the isoclasses of class 2 groups of exponent $p$ and order $p^{m+n}$ and with derived subgroup ${\mathbb{Z}}_p^n$, and (ii) the set $\text{Gr}(n,{\text{AS}}_m({\mathbb{Z}}_p))/{\text{GL}}_m({\mathbb{Z}}_p)$ of orbits of $\text{Gr}(n,{\text{AS}}_m({\mathbb{Z}}_p))$ under the congruence action by ${\text{GL}}_m({\mathbb{Z}}_p)$, where $\text{Gr}(n,{\text{AS}}_m({\mathbb{Z}}_p))$ is the set of $n$-dimensional subspaces of anti-symmetric matrices of order $m$ over ${\mathbb{Z}}_p$. We give a description of the orbit spaces $\text{Gr}(2,{\text{AS}}_m({\mathbb{Z}}_p))/{\text{GL}}_m({\mathbb{Z}}_p)$ for all $m$ and $p$ by applying the theory of pencils of anti-symmetric matrices. Based on this, we show complete sets of representatives of orbits of $\text{Gr}(3,{\text{AS}}_4({\mathbb{Z}}_3))/{\text{GL}}_4({\mathbb{Z}}_3)$, $\text{Gr}(4,{\text{AS}}_4({\mathbb{Z}}_3))/{\text{GL}}_4({\mathbb{Z}}_3)$ and $\text{Gr}(3,{\text{AS}}_5({\mathbb{Z}}_3))/{\text{GL}}_5({\mathbb{Z}}_3)$. As a consequence, we obtain a classification of corresponding class 2 groups of exponent $p$. In particular, we recover the classification of groups with exponent 3 and order $\le 3^8$.