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 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.

References

Journal references

• Non-PORC behaviour of a class of descendant p-groups by Marcus du Sautoy and M. R. Vaughan-Lee, Journal of Algebra, ISSN 00218693, Volume 361, Page 287 - 312(Year 2012): ^{Gated copy (PDF)More info}