Coclass

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Mathematical concept

title: "Coclass" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["p-groups"] description: "Mathematical concept" topic_path: "general/p-groups" source: "https://en.wikipedia.org/wiki/Coclass" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept ::

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

The coclass conjectures

The coclass conjectures were introduced by and proved by and . 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.

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}}

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p-groups