Descendant tree (group theory)

In the realm of group theory, a descendant tree provides a visual and structural representation of pro-p groups and their finite quotients, especially p-groups. It's a hierarchical organization depicting the relationships between different groups obtained by successively applying p-coverings or similar constructions. It’s particularly useful for exploring the structure and classification of p-groups.

The root of the descendant tree is typically a relatively small p-group, often with a simple structure. Subsequent levels are populated by its descendants. A descendant of a p-group G is another p-group H such that there exists a central extension 1 → A → H → G → 1, where A is an elementary abelian p-group (i.e., a vector space over the field with p elements) contained in the Frattini subgroup of H. This means that G is a quotient of H by a central subgroup of exponent p that is contained in all maximal subgroups of H.

Each node in the tree represents a specific p-group, and the edges connecting nodes represent the descendant relationship. The number of children a node has is determined by the number of distinct central extensions (subject to some equivalence relation, typically isomorphism) of the p-group it represents. The branching pattern of the tree reveals information about the prevalence and diversity of p-groups with specific properties.

Key concepts related to descendant trees include:

• Immediate Descendant: An immediate descendant of a group G is a descendant H obtained by a single central extension as described above. An edge in the descendant tree connects a group to its immediate descendants.

• Terminal Node: A node in the tree representing a group that has no further descendants (according to the specific criteria for constructing the tree). These groups often have unique properties or represent "end points" in the classification of p-groups.

• Coclass: The coclass of a finite p-group G of order pⁿ and nilpotency class c is defined as n - c. Coclass is often used to organize and analyze p-groups. Descendant trees are frequently structured by coclass, allowing researchers to investigate the structure of p-groups with a particular coclass.

• Pro-p Group: A pro-p group is a topological group that is an inverse limit of finite p-groups. Descendant trees are instrumental in studying the structure of pro-p groups, as they provide a way to systematically explore the finite quotients of a pro-p group. The infinite pro-p groups can be "approximated" by examining increasingly deeper levels of their descendant tree.

Descendant trees are used extensively in the classification of p-groups, especially for specific coclasses. They are powerful tools for visualizing and understanding the intricate relationships between different p-groups and the underlying structure of pro-p groups.