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Define a relation on G

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Reference no: EM131780121

Q1: Prove that if x2 =1 for all x ∈ G  then G is abelian.

Q2: Prove that A x B is an abelian group if and only if both and are abelian.

Q3: Let be an abelian group and fix some n ∈ z. Prove that the set {an|a ∈ A} is a subgroup of A.

Q4: Let H ≤ G and define a relation ~ on G by a~b if and only if b-1a ∈ H.

Prove that ~ is an equivalence relation.

Q5: Prove that if H and K are finite subgroups of G whose orders are relatively prime then H ∩ K = 1.

Reference no: EM131780121

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