Constructing CF groups by coclass. (English) Zbl 1187.20022

Eick, Bettina; Feichtenschlager, Dörte

Exp. Math. 18, No. 2, 205-211 (2009).

According to N. Blackburn [Acta Math. 100, 45-92 (1958; Zbl 0083.24802)], a finite \(p\)-group \(G\) is said to be a CF-group if the quotients of the lower central series, except the first one, have order at most \(p\), that is, \(|\gamma_i(G)/\gamma_{i+1}(G)|\leq p\) for \(i\geq 2\). A CF-group \(G\) is said to be an ECF-group if \(G/\gamma_2(G)\) is elementary Abelian. The ECF-groups with \(G/\gamma_2(G)\) of order \(p^2\) are the \(p\)-groups of maximal class, which have been widely studied [see the book by C. R. Leedham-Green and S. McKay, The structure of groups of prime power order. (Lond. Math. Soc. Monographs. New Series 27. Oxford: Oxford University Press (2002; Zbl 1008.20001))].
The paper under review lays the foundations of a theory of CF-groups, from the point of view of pro-\(p\)-groups and coclass theory. A class of infinite CF pro-\(p\)-groups with fixed coclass is constructed. It is shown that this implies that CF pro-\(p\)-groups form a rather large collection within the collection of all just-infinite pro-\(p\)-groups of fixed coclass.

Reviewer: A. Caranti (Trento)

MSC:

20E18	Limits, profinite groups
20D15	Finite nilpotent groups, \(p\)-groups
20F18	Nilpotent groups

Keywords:

\(p\)-groups; coclass; CF groups; \(p\)-adic space groups; pro-\(p\)-groups

Citations:

Zbl 0083.24802; Zbl 1008.20001

Software:

SymbCompCC