 # Abstract Algebra Problems

## Question

1. If N is a subgroup of index 2 in a group G, then N is normal in G.

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.

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