Immediate descendant

Definition
Suppose $$G$$ is a finite p-group. A finite p-group $$K$$ is termed an immediate descendant of $$G$$ if $$G \cong K/\lambda_c(K)$$ where $$c$$ is the defining ingredient::exponent-p class of $$K$$ and $$\lambda_c(K)$$ denotes the $$c^{th}$$ member of the lower exponent-p central series of $$K$$.

Note in particular that the value of $$c$$ is the same for all groups having $$G$$ as immediate descendant, and all these are one more than the exponent-p class of $$G$$ itself.

Every finite $$p$$-group that is not an elementary abelian $$p$$-group arises as the immediate descendant of some nontrivial finite $$p$$-group. Elementary abelian $$p$$-groups are anomalous in that they can be thought of as immediate descendants of the trivial group.