Let's analyze each exercise one by one.

Problem E7

We are given a set $G$ of matrices: $G = \left\{ A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \;|\; a, b \in \mathbb{R}, \det(A) \neq 0 \right\}.$ We need to prove that $G$ forms a group under matrix multiplication. To do this, we must show that:

1. Closure: The product of any two matrices in $G$ is also in $G$.
2. Associativity: Matrix multiplication is associative.
3. Identity Element: There exists an identity element in $G$.
4. Inverse Element: For each matrix in $G$, there exists an inverse in $G$.

Problem E8

We have a set of matrices: $\mathcal{M} = \left\{ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}^n \;|\; n \in \mathbb{N}, n \geq 1 \right\} \subset M_3(\mathbb{C}).$

Part (a)

We need to prove that $(\mathcal{M}, \cdot)$ is a commutative group, meaning that it satisfies the four group properties and is commutative.

Part (b)

We are asked to study whether the operation $A \circ B = A^4 \cdot B^4$, defined on $\mathcal{M}$, gives it a group structure.

Problem E9

We are given the set $A_n = \{ \sigma \in S_n \;|\; \sigma \text{ is an even permutation} \}$.

Part (a)

We need to prove that $(A_n, \circ)$ is a group. This means verifying the group properties with the set of even permutations under composition.

Part (b)

We need to determine for which values of $n$ the group $A_n$ is commutative.

Would you like a detailed solution for each part? Here are some further questions for exploration:

1. What are the specific elements in $G$ that might fail the group properties?
2. How can we verify closure under matrix multiplication in $G$?
3. What are the implications of the operation $A \circ B = A^4 \cdot B^4$ for the set $\mathcal{M}$?
4. How do we determine if an even permutation set $A_n$ is commutative for specific values of $n$?
5. What properties distinguish even and odd permutations in symmetric groups?

Tip: When proving group properties, carefully check closure and inverses as they are commonly tricky in matrix and permutation groups.