Exponent-p class

Definition

Suppose $p$ is a prime number and $G$ is a nilpotent p-group that is also a group of finite exponent (note that any finite p-group would satisfy these conditions, but there are also some infinite $p$-groups that satisfy the two conditions). The exponent-$p$ class of $G$, also called the $p$-class of $G$ or the Frattini class of $G$, is defined in the following equivalent ways:

1. It is the length of the lower exponent-p central series of $G$.
2. It is the length of the socle series of $G$.
3. It is the minimum possible length of an exponent-p central series of $G$.