Abstract Algebra Answers

2.10. Let H be the subgroup generated by two elements a, b of a group G. Prove that if ab = ba, then H is an abelian group.

2.9. Let a and b be integers.

(a) Prove that the subset aZ + bZ = {ak + bl | l, k ∈ Z } is a subgroup of Z.

(b) Prove that a and b + 7a generate the subgroup aZ + bZ.

2.8. Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

2.7. If G is a group such that (ab)2 = a2b2 for all a, b ∈ G, then show that G must be abelian.

2.6. If G is a group in which (ab)ⁱ = aⁱbⁱ for three consecutive integers i for all a, b ∈ G, show that G is abelian.

2.5. If G is a finite group, show that there exists a positive integer m such that a^m = e for all a ∈ G.

2.4. If G is a group of even order, prove that it has an element "a\\ne e" satisfying a² = e.

2.3. Let G be a nonempty set closed under an associative product, which in addition satisfies:

(a) There exists an e ∈ G such that ae = a for all a ∈ G.

(b) Given a ∈ G, there exists an element y(a) ∈ G such that ay(a) = e.

Prove that G must be a group under this product.

Assume that the equation xyz = 1 holds in a group G. Does

it follow that yzx = 1? That yxz = 1? Justify your answer.

2.1. Let S be any set. Prove that the law of multiplication defined

by ab = a is associative.