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).


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
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[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
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