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Finite Groups whose Powers have no Countably Infinite Factor Groups

Published online by Cambridge University Press:  20 November 2018

M. J. Billis
M. J. Billis*
Affiliation:
University of Utah, Salt Lake City, Utah Montana State University, Bozeman, Montana
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 965 - 969
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Neumann, B. H. and Yamamuro, S., Boolean powers of simple groups, J. Austral. Math. Soc. 5 (1965), 315–324.Google Scholar
2. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, N.J., 1964).Google Scholar
3. Wielandt, H., Ein Veralgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), 209–244.Google Scholar