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31 of order Abelian arbitrary element ascending central series assertion assumption characteristic subgroup commutes composition factors composition series condition conjugate in 31 Consider the normalizer contains contradiction direct product divide the order divis divisible by q element of 31 element of order exist groups finite group fore group Bg group is p-soluble group of order groups whose order Hence it follows highest power II-separable II-Sy II-Sylow II-Sylow-divisor II-Sylow-regular insoluble isomorphic l(modp least one soluble lemma less than g Let us assume nilpotent groups normal sub number with respect obtain obviously order g order not divisible order of _ p-Abelian group p-commutator p-decomposable groups p-derived group p-nilpotent p-Sylow p-Sylow-subgroup prime divisor principal series properties satisfies series of normal smallest order soluble and conjugate soluble with respect subgroup of order supersoluble with respect Sylow's theorem Theo theorem holds Theorem III theorem is false theorem of Burnside theorem of Sylow Theorem VIII