QuestionQuestion

Prove or disapprove the following:

1. Let G be a group and let H be a subgroup of G.
If N(H) = {x ∈ G | ∀ h ∈ H, xh⁻¹ ∈ H}, then N(H) is a subgroup of G.

2. If G is a group with exactly 8 elements or order 10, then it has exactly 8 cyclic subgroups of order 10.

3. Let H = {g² | g ∈ G} for some group G. Then H ≤ G.

4. There exists a nonabelian group of order 4.

5. For isomorphism Φ: G --> H and a, b ∈ G. If a⁻¹ = b, then Φ(a)⁻¹ = Φ(b).

6. All cyclic groups are abelian.

7. All abelian groups are cyclic.

8. D₄ contains no cyclic subgroups.

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Abstract Algebra Proofs
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