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FACT0R GR0UPS PERMUTATI0N REPRESENTATI0NS
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0rder a2b a3b a"b a3b a2b axis bc bac called cb cb2 composition series conjugacy classes conjugate consider cyclic subgroup DEFINITI0N denote direct product direct sum element of order elements of G exercise factor group finite group G is abelian G of order given group G group of order group theory H is normal Hence G homomorphism identity infinite cyclic group inverse Klein 4-group Lagrange's theorem lattice of subgroups left cosets Let f Let G Let H matrix multiplication table non-abelian group normal in G normal subgroup number of elements obtained one-one order p2 positive integer possible groups PR00F Let prime proof result right cosets simple groups solvable group solvable series structure theory subgroup H subgroup of G subgroups of order Subgroups order subnormal series subset Suppose Sylow p-subgroup Sylow theorems Sylow's third theorem symmetry group theorem follows x~lAx