Supersoluble Immersion

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Abstract
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: (i)K is supersolubly immersed in G. (ii)K/ϕK is supersolubly immersed in G/ϕK. (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.

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