Coclass

In mathematics, the coclass of a finite p-group of order pⁿ is n − c, where c is the nilpotency 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 solvable pro p-groups of given coclass.

Leedham-Green, C. R.; Newman, M. F. (1980), "Space groups and groups of prime-power order. I.", Arch. Math., 35 (3), Basel: 193–202, doi:10.1007/BF01235338, MR 0583590
Leedham-Green, C. R. (1994), "The structure of finite p-groups", J. London Math. Soc., Series 2, 50 (1): 49–67, doi:10.1112/jlms/50.1.49, MR 1277754
Shalev, Aner (1994), "The structure of finite p-groups: effective proof of the coclass conjectures", Invent. Math., 115 (2): 315–345, doi:10.1007/bf01231763, MR 1258908