## On ABA-groups of finite order |

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4,Theorem A-indecomposable A-invariant subgroup A-irreducible a'eA a^eA ABA-groups abelian Hall subgroup acts irreducibly acts regularly Assume false b^eB beB-BQ beBQ bO bO central extension CG(a char Clifford's Theorem completes the proof contradiction completes da=d divide G elementary abelian 2-group exists an element exists an involution factor group false and let finite group follov;s following conditions hold follows from 6.4 follows immediately Furthermore G containing G is factorizable G is solvable G possesses group of G groups of type hence A=A Hence G iant inverse image inverts the elements involution in Bq isomorphic Lemma Let aeA Let G=ABA maximal subgroup minimal order N=ABQ Ng(A nilpotent non-trivial number of conjugates odd order odd prime order q-1 particular possesses a normal Proof of vii proper subgroup Proposition solvable group subgroup of G suitable integer Suppose there exists Sylow 2-subgroup T=DA Theorem thesis trivial