# Word automatic groups of nilpotency class 2

## TL;DR

This paper investigates the word automaticity of certain nilpotent groups of class 2 with prime exponent, demonstrating that some are automatic while others are not, and introduces a method for showing automaticity of central extensions.

## Contribution

It establishes which nilpotent class 2 groups with prime exponent are word automatic and introduces a new method for proving automaticity of central extensions.

## Key findings

- The infinitely generated free group of this variety is not word automatic.
- The infinite extra-special p-group E_p is word automatic.
- A new method using co-cycles is introduced for automaticity of central extensions.

## Abstract

We consider word automaticity for groups that are nilpotent of class $2$ and have exponent a prime $p$. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special $p$-group $E_p$ is word automatic, as well as an intermediate group $H_p$ which has an infinite centre. In the last section we introduce a method to show automaticity of central extensions of abelian groups via co-cycles.