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.

## See also

* Descendant tree (group theory)

## References

* "Space groups and groups of prime-power order. I.", Arch. Math. (Basel) 35 (3): 193–202, 1980, doi:10.1007/BF01235338
* "The structure of finite p-groups", J. London Math. Soc., Series 2 50 (1): 49–67, 1994, doi:10.1112/jlms/50.1.49
* 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

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