2. Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G.
3. If H is a normal subgroup of a group G such that H and G/H is finitely generated, then so is G.
4. Let N ∇ G and K ∇ G. If N ∩ K = 〈e〉 and N ∨ K = G, then G/N ≅ K.
5. If ƒ : G → H is a homomorphism, H is abelian and N is a subgroup of G containing Ker ƒ, then N is normal in G.
6. If N ∇ G, |N| finite, H < G, [G : H] finite, and [G : H] and |N| are relatively prime, then N < H.
7. Let G be a group and C be its center. Prove that if G/C is a cyclic group, then G is an abelian group.
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.
By purchasing this solution you'll be able to access the following files: