Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). An abelian group of order 81 is a cyclic group. (2). Suppose 9₁ and 92 are two cycles in S100. If the length of cycles 9₁ and 92 both acts transitively on {1, 2, ..., 100 }, then there exists an element h E S100 such that hg₁h-1 = 92. (3). and H₂ in G are finite and coprime to each other. Then G/H₁ H₂ (G/H₁) × (G/H₂). Let H₁ and H₂ be two normal subgroups of G. Suppose the indexes of H₁

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