Chapter 6.2, Problem 1E

Show that each of the following algebraic structures is a group. Which groups are abelian?

1. $\left(\left\{1,-1\right\},\cdot \right),$ where $\cdot$ is integer multiplication.
2. $\left(\left\{1,\alpha ,\beta \right\},\cdot \right),$ where $\alpha =\frac{-1+i\sqrt{3}}{2},\beta =\frac{-1-i\sqrt{3}}{2},$ and $\cdot$ is complex number multiplication.,
3. $\left(\left\{1,-1,i,-i\right\},\cdot \right),$ where $\cdot$ is complex number multiplication.
4. $\left(P\left(X\right),\Delta \right),$ where X is a nonempty set and $\Delta$ is the symmetric difference operation $A\Delta B=\left(A-B\right)\cup \left(B-A\right).$
5. $\left(t\mathbb{Z},+\right),$ where t is a natural number.
6. The set of $2\times 2$ real matrices with determinant 1, where the operation is matrix multiplication.