Coclass

Updated on Jan 08, 2024

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In mathematics, the coclass of a finite p-group of order pⁿ is n − c, where c is the class.

The coclass conjectures

The coclass conjectures were introduced by Leedham-Green and Newman (1980) and proved by Leedham-Green (1994) and Shalev (1994). They are:

Conjecture A: Every p-group has a normal subgroup of class 2 with index depending only on p and its coclass.

Conjecture B: The solvable length of a p-group can be bounded in terms of p and the coclass.

Conjecture C: A pro p-group of finite coclass is solvable.

Conjecture D: There are only finitely many pro p-groups of given coclass.

Conjecture E: There are only finitely many soluble pro p-group of given coclass.

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