coclass
{{Short description|Mathematical concept}}
{{For|the gamebird species|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 {{harvs|txt|last1=Leedham-Green|last2= Newman |year=1980}} and proved by {{harvs|txt|last=Leedham-Green|year=1994}} and {{harvs|txt|last=Shalev|year=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
References
- {{citation|mr=0583590
|last=Leedham-Green|first= C. R.|author-link1=Charles Leedham-Green |last2= Newman|first2= M. F.
|title=Space groups and groups of prime-power order. I.
|journal=Arch. Math. |place=Basel|volume= 35 |year=1980|issue= 3|pages= 193–202|doi=10.1007/BF01235338}}
- {{citation|mr=1277754
|last=Leedham-Green|first= C. R. |author-link=Charles Leedham-Green
|title=The structure of finite p-groups
|journal=J. London Math. Soc. |series=Series 2|volume= 50 |year=1994|issue=1|pages= 49–67|doi=10.1112/jlms/50.1.49|doi-access=free}}
- {{citation|mr=1258908
|last=Shalev|first= Aner
|title=The structure of finite p-groups: effective proof of the coclass conjectures
|journal=Invent. Math.|volume= 115 |year=1994|issue= 2|pages= 315–345|doi=10.1007/bf01231763}}