Descendant tree (group theory)

In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order ${\displaystyle p^{n}}$, for a fixed prime number ${\displaystyle p}$ and varying integer exponents ${\displaystyle n\geq 0}$. Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups.

Additionally to their order ${\displaystyle p^{n}}$, finite p-groups have two further related invariants, the nilpotency class ${\displaystyle c}$ and the coclass ${\displaystyle r=n-c}$. It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass ${\displaystyle r}$, reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized presentations. Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure.

An important question is how the descendant tree ${\displaystyle {\mathcal {T}}(R)}$ can actually be constructed for an assigned starting group which is taken as the root ${\displaystyle R}$ of the tree. The p-group generation algorithm is a recursive process for constructing the descendant tree of a given finite p-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma.