Coclass

For the gamebird species, see Koklass pheasant.

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.

See also

• Descendant tree (group theory)