On Monomial Groups |
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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 C₁ C2 X C2 characteristic subgroup commutator subgroup Consider contradiction Crelle's Journal cyclic group Definition dihedral group Dornhoff element elementary abelian factor group follows 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 monomial group monomiale groepen nilpotent groups non-abelian non-monomial groups normal in G order 24 order 48 order 9 p-group is modular proof of Th prove quaternion group representation groups representation of G S₁ S₂ saturated formation Satz Schur solvable group subgroup of G subgroup of order sufficient completed group Suppose Sylow p-subgroup Sylow's theorem V₁ W₁ W₂ welke whence