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0-immersion is factor 5-dispersed abelian group action of G Apply Assume belongs characteristic subgroup class of finite conics conjugate classes contained contradiction cyclic group deduce defined denote divisor of e(p element elementary abelian p-group epimorphic image epimorphism of G exists a maximal factor inherited factor of G finite group following properties Frattini subgroup functions G are equivalent G upon H g-group group G group of automorphisms H with M C H/cB Hence hypercentral ii.b immersion implies induces irreducible characters isomorphic Lemma linear lower defect groups Math maximal subgroup mB(P minimal normal subgroup Modular Law morphism multiple nilpotent nilpotent group normal sub p-element p-subgroup pair 21 permutation permutation group primary elementary abelian primary subgroup principal factor Proposition 1.5 Proposition 4.10 proving Qt(G quadrics r a divisor representation shows soluble stabiliser subgroup of G subset supersoluble symmetric group Theorem 5.3 validity