 # Abstract Algebra Proofs

## Question

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.

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden. \$25.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Abstract Algebra Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.