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Embedding theorems in finite soluble groups Hughes, Peter Walter


By a group we will mean a finite soluble group. It is an interesting fact, (Pardoe [1]), that the subgroup closure of the class of groups P[symbol omitted], those with a unique complemented chief series, is all groups. Let X be the class of groups with a complemented chief series. We investigate the action of closure operations T such that TX = X upon P[symbol omitted]. The purpose of this is to find a collection of such closure operations whose join applied to P[symbol omitted] is X . In the course of this investigation we introduce a new closure operation M defined by; MY = { G | G = <X₁,•••,Xn>, X₁ ɛ Y, X₁ sn G, ( |G : X₁|,•••,|G : Xn| ) = 1 }

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