A characterization of groups of exponent $p$ which are nilpotent of class at most 2

Abstract

Let $(\mathbf{G},+)$ be a group of prime exponent $p = 2n + 1$. In this paper we prove that $(\mathbf{G},+)$ is nilpotent of class at most 2 if and only if one of the following properties is true: 
$i)$ $\mathbf{G}$ is also the support of a commutative group $(\mathbf{G},+')$ such that $(\mathbf{G},+)$ and $(\mathbf{G},+')$ have the same cyclic cosets [cosets of order $p$].     $ii)$ the operation $\oplus$ defined on $\mathbf{G}$ by putting $x \oplus y = x/2 + y + x/2$, gives $\mathbf{G}$ a structure of commutative group.\end