On nilpotent groups and conjugacy classes

Abstract

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime number. Set $a^G b^G=\{xy\mid x\in a^G, y\in b^G\}$. Then either $a^G b^G=(ab)^G$ or $a^G b^G$ is the union of at least $\frac{p+1}{2}$ distinct conjugacy classes. As an application of the previous result, given any nilpotent group $G$ and any conjugacy class $a^G$ of size $p$, we describe the square $a^G a^G$ of $a^G$ in terms of conjugacy classes of $G$.