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MULTIPLICATION TABLE GENERATORS
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3—cycle abelian groups axis azb a3b azbz bzac C2 X C2 called composition series conjugacy classes conjugate consider cyclic subgroup defined denote direct product direct sum disjoint element of order elements of G exercise factor group finite group finite point groups function f given group G group of order group theory H is normal Hence G homomorphism identity improper rotation infinite cyclic group inverse isomorphic Kerf Klein 4—group Lagrange's theorem lattice of subgroups left cosets Let f Let G Let H multiplication table non-abelian group normal in G normal subgroup number of elements one—one orbit permutation group positive integer prime PROOF Let result right cosets simple groups solvable group solvable series subgroup H subgroup of G subgroups of order Subgroups order subnormal series subset Suppose Sylow p-subgroup Sylow theorems symmetry group