Suppose that H is a subgroup of a finite group G and that |H| and(|G:H| - 1)! are relatively prime. Prove that H is normal in G. Whatdoes this tell you about a subgroup of index 2 in a finite group?

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Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

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A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

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A: This is a problem of Group Theory.

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Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

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A: Since you have asked multiple question, we will solve the first question for you. If you want any…