Recall that $GL_{2}(\mathbb{R})$ denotes the group of 2x2 invertible matrices with real entries with the product given by matrix multiplication. Let H denote the smallest subgroup of $GL_{2}(\mathbb{R})$ containing:
$A = (\begin{smallmatrix} 1&0\\ 0&-1 \end{smallmatrix})$ $B = \frac{1}{2}(\begin{smallmatrix} -1&-\sqrt{3}\\ \sqrt{3}&-1 \end{smallmatrix})$.
i) What is the order of A? Order of B? Compute $BA$ and $AB^{2}$. ii) What is the order of H? Is H isomorphic to $\sum_{3}$? If you think H and $\sum_{3}$ are isomorphic, give an explicit isomorphism. Hint: $sin(2\pi/3)= \sqrt{3}/2$.
Ive done i and have the order of A as 2, the order of B as 3. $$BA = (\begin{smallmatrix} \frac{-1}{2}&\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&\frac{-1}{2} \end{smallmatrix})$$
$$AB^{2} = (\begin{smallmatrix} \frac{-1}{2}&\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&\frac{-1}{2} \end{smallmatrix})$$
I am just struggling with ii, any help please?
Hint: We can interpret $A$ as a reflection over some axis and we can interpret $B$ as a rotation by $1/3$ of a full turn. So $H$ is secretly the dihedral group of a triangle: $$ H = \{I_2, B, B^2, A, AB, AB^2\} $$ It remains to show that the dihedral group of a triangle is isomorphic to the symmetric group on three elements.