A note on some special $p$-groups

TL;DR

This paper investigates the structure and properties of special p-groups, providing bounds and explicit descriptions of their Schur multipliers, exterior, and tensor squares, especially for groups with maximal or near-maximal derived subgroup order.

Contribution

It extends previous bounds on Schur multipliers of special p-groups and characterizes their structure and capability for specific derived subgroup sizes.

Findings

Derived bounds for Schur multipliers, exterior, and tensor squares.

Explicit structure description for groups of exponent p.

Identification of capable groups among special p-groups.

Abstract

Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\frac{1}{2}d(d-1)-1}$ for $ d\geq 3 $ and $ p\neq 2. $ Additionally, when $ G $ is of exponent $ p, $ we give the structure of $ G. $