In mathematics, the coclass of a finite
''p''-group of order ''p''
''n'' is ''n'' − ''c'', where ''c'' is the class.
The coclass conjectures
The coclass
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
s were introduced by and proved by and . They are:
*Conjecture A: Every
''p''-group has a normal
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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) In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order p^n, for a fixed prime number p and varying intege ...
References
*
*
*{{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
P-groups