# Free product of class two of two elementary abelian groups of prime-square order

View other such prime-parametrized groups

## Definition

Let $p$ be a prime number. This group is defined as the quotient of the free product of two copies of the elementary abelian group of prime-square order with prime $p$, by the third member of its lower central series. In other words, it is the free product in the sense of groups of nilpotency class two of two copies of the elementary abelian subgroup of order $p^2$.

An explicit presentation is given as follows. If we define:

$K := \langle a_1, a_2, b_1, b_2 \mid a_1a_2 = a_2a_1, b_1b_2 = b_2b_1, a_1^p = a_2^p = b_1^p = b_2^p = e\rangle$

Then our group is:

$G := K/[[K,K],K]$

(this can be written down in a single presentation by writing all the nontrivial commutators $[[x,y],z]$ with $x,y,z \in \{a_1,a_2,b_1,b_2\}$.