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Supersoluble Immersion

Published online by Cambridge University Press:  20 November 2018

Reinhold Baer
Reinhold Baer*
Affiliation:
Frankfurt am Main
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Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:

If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).

Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:

  1. (i) K is supersolubly immersed in G.

  2. (ii) K/ϕK is supersolubly immersed in G/ϕK.

  3. (iii) If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.

Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 353 - 369
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Baer, R., Supersoluble groups, Proc. Amer. Math. Soc, 6 (1955), 16–32.Google Scholar
2. Baer, R., Classes of finite groups and their properties, Illinois Jour. Math.,.1 (1957), 115–187.Google Scholar
3. Baer, R., Closure and dispersion, Illinois Jour. Math., 3 (1959), 000-000.Google Scholar
4. Hall, P., A contribution to the theory of groups of prime power order, Proc. Lond. Math. Soc, 36 (1932), 29–95.Google Scholar
5. Huppert, B., Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Zeit., 60 (1954), 409-34.Google Scholar
6. Jacobson, N., The Theory of Rings (New York, 1943).Google Scholar
7. Zassenhaus, H., Lehrbuch der Gruppentheorie, I (Leipzig und Berlin, 1937).Google Scholar