Geodesic
Finite Groups whose Powers have no Countably Infinite Factor Groups
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 965-969

Voir la notice de l'article provenant de la source Cambridge University Press

Let P be the class of all finite groups G whose powers GI have no countably infinite factor groups. Neumann and Yamamuro (1) proved that if G is a finite non-Abelian simple group, then G ∈ P. We generalize this result by proving the following theorem.THEOREM. A finite group G ∈ P if and only if G is perfect 2. Inheritance properties ofP.P1. If G ∈ P and N is normal in G, then G/N ∈ P. Proof. Since (G/N)I is isomorphic to GI/NI it is clear that factor groups of (G/N)I are isomorphic to factor groups of GI , and hence finite or uncountable.P2. If G = HK, where H ∈ P and K ∈ P, then G ∈ P. Proof. We show that homomorphic images of GI are either finite or uncountable. Let φ be a homomorphism of GI. Then GIφ = (HK)Iφ = (HI/KI)φ = (HIφ) (KIφ). Since HIφ and KIφ must be finite or uncountable, the conclusion follows. 
Billis, M. J. Finite Groups whose Powers have no Countably Infinite Factor Groups. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 965-969. doi: 10.4153/CJM-1969-105-1
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[1] 1. Neumann, B. H. and Yamamuro, S., Boolean powers of simple groups, J. Austral. Math. Soc. 5 (1965), 315–324. Google Scholar

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[3] 3. Wielandt, H., Ein Veralgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), 209–244. Google Scholar

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