Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then GH is cyclic. Definition: A subgroup H of a group is said to be a normal subgroup of G it for all a∈G, aH = Ha Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set GH with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.

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Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then GH is cyclic.  Definition: A subgroup H of a group is said to be a normal subgroup of G it for all a∈G, aH = Ha  Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set GH with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.

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Let G be a group, and H be a normal subgroup of G.  Assume that G is cyclic.  To prove: GH is cyclic.  Proof: As H is normal and G is cyclic, then G is Abelian.  Let us suppose,   G=⟨a⟩={ai∣i∈Z}  By definition of GH, we have  GH={xH∣x∈G}  Use the hypothesis that G is cyclic, hence it gives  GH={xH∣x∈G}={aiH∣ai∈G}={aH)i∣ai∈G}=⟨aH⟩  Hence, it is proved that GH is cyclic.

Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then G/H is cyclic. Definition: A subgroup H

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