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 $$G$$.