×

zbMATH — the first resource for mathematics

Free pro-\(p\) groups with operators. (English) Zbl 0755.20006
A semidirect product of a finite \(p\)-group \(P\) by a finite group \(H\) such that the induced representation of \(H\) on the Frattini factorgroup of \(P\) is projective can be extended to the corresponding product of a free pro- \(p\)-group by \(H\) (theorem A). Finitely generated free pro-\(p\)-groups on which \(H\) acts such that the Frattini factorgroups are projective \(\mathbb{F}_ pH\)-modules are projective objects in the category of finitely generated pro-\(p\)-groups with operator group \(H\) (theorem B).
MSC:
20E18 Limits, profinite groups
20E22 Extensions, wreath products, and other compositions of groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20C25 Projective representations and multipliers
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Baumann, B.: Über die Struktur von p-Normalteilern endlicher Gruppen. J. London Math. Soc. (2), 28(1983), 477–480 · Zbl 0546.20016
[2] Baumann, B.: Erweiterungen linearer Gruppen mit p-Gruppen. Arch. Math. 55 (1990), 317–323 · Zbl 0679.20027
[3] Gruenberg, K.W.: Cohomological topics in group theory. Springer Lect. Notes 143, (1970) · Zbl 0205.32701
[4] Humphreys, J.E.: The Steinberg representation. Bull. Am. Math. Soc. 16, (1987), 247–263 · Zbl 0627.20024
[5] Huppert, B.: Endliche Gruppen I, Springer-Verlag (1967) · Zbl 0217.07201
[6] Kraus, G.: Erweiterungen linearer Gruppen mit 2-Gruppen. Diplomarbeit, Giessen (1988/89)
[7] Kurosh, A.G.: The theory of groups I, II. Chelsea Publishing Company (1960) · Zbl 0094.24501
[8] Landrock, P.: Finite group algebras and their modules. LMS Lecture Note Series 84, Cambridge University Press (1983) · Zbl 0523.20001
[9] Lubotzky, A.: Combinatorial group theory for pro-p groups. J. Pure and Appl. Algebra 25 (1982), 311–325 · Zbl 0489.20024
[10] Mostowski, A.W.: On automorphisms of relatively free groups. Fund. Math. 50, (1961/62), 403–411 · Zbl 0105.02002
[11] Neumann, H. Varieties of groups. Ergeb. Math. 37, Springer (1967) · Zbl 0149.26704
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.