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- W3013570370 abstract "Let $p$ be a prime number and $mathbb{Z}_p=mathbb{Z}/pmathbb{Z}$. 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 $le 3^8$." @default.
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- W3013570370 date "2020-03-28" @default.
- W3013570370 modified "2023-09-27" @default.
- W3013570370 title "Extension of elementary $p$-groups and its application in classification of groups of prime exponent" @default.
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