## Classes of Conjugate Subgroups in Finite Solvable Groups |

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a d 9-subgroup assume Carter subgroup central chief factor class of conjugate class of groups complemented in G conjugate in G contained in f containing each group contains a minimal contains a subgroup contains each subgroup contradiction proves Definition epimorphic image exactly one minimal f subgroup factor of G Fitting property Frattini argument Frattini chief factor G contains G is contained G satisfying G then G group G group of minimal Hence G KN/LN Lemma Lemma 2.10 implies Let F Let G Let K/L maximal subgroup minimal normal subgroup minimality of G Mq/M nilpotent group non-central chief factor normalizer of G Nq(D NQ(E Nq(F p-chief factor K/L p-group p-subgroup Proof Proposition 2.9 ptp(G satisfies the hypothesis saturated formation set f(p solvable groups statement is false subgroup F subgroup of G supersolvable groups Sylow system system normalizer system of G tained U-chief unique minimal normal