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1-dimensional representation abelian normal subgroup abelian of order abelian Sylow subgroups algebraically closed field Artin L-functions Burnside's basis theorem C Z(P C2 x C2 characteristic subgroup commutator subgroup Consider contradiction cyclic group Definition dihedral group Dornhoff element elementary abelian exists factor group Frattini subgroup G is monomial geldt groep group G groups of order hence G homomorphic image Huppert Huppert's theorem hypersolvable groups hypersolvable of order induces an automorphism irreducible representation isomorphic Lemma Let G Main Theorem modular p-group module monomial by Th monomial group monomiale groepen N C Z(G nilpotent groups NM/M non-monomial groups normal in G order 2k order 9 p-group is modular proof of Th prove quaternion group representation groups representation of G saturated formation Satz Schur SL F SL_(F solvable group subgroup of G subgroup of order sufficient completed group Suppose Sylow p-subgroup Sylow's theorem unique normal subgroup welke whence zijn