For a given set of matrices, how can i verify that they form a group under multiplication? I am looking for specific conditions so that i can insert them in a code to verify whether any set of matrices forms a group under multiplication.
Exemple: ''' The multiplicative group of Weyl $\mathcal{G}_{w2}$ has 12 cosets $\{C_0,C_2\dots,C_{11}\}$, each one containing 16 unitary (hence invertible) matrices, where the first coset $C_0$ is defined as: \begin{equation*} C_0= \left \{ \alpha \begin{bmatrix}1 & 0 \\ 0 & \pm 1 \end{bmatrix} \ , \ \alpha \begin{bmatrix}0 & 1 \\ \pm1 & 0 \end{bmatrix} \right \} \end{equation*} with $\alpha \in \{ +1,-1,+i,-i \}$. All the 12 cosets are derived from $C_0$ as follows: \begin{equation*} C_k = a_k C_0 \ \forall k=0,1,\dots ,11 \end{equation*} where $a_k$ can be grouped into two sets, the first one being $\{a_0,a_1,\dots,a_5\}$ where: \begin{align*} a_0 = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \ a_1 = \begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix} \ a_2 = {\frac{1}{\sqrt{2}}} \begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix}\\ a_3 = {\frac{1}{\sqrt{2}}} \begin{bmatrix}1 & 1 \\ i & -i \end{bmatrix}\ a_4 = {\frac{1}{\sqrt{2}}} \begin{bmatrix}1 & i \\ 1 & -i \end{bmatrix} \ a_5 = {\frac{1}{\sqrt{2}}} \begin{bmatrix}1 & i \\ i & 1 \end{bmatrix} \end{align*} The second set $\{a_6,a_7,\dots a_{11}\}$ is given by: $a_{k+6}= \eta a_k$, with $\eta =(1+i)/\sqrt{2} \ \ \forall k =0,1,\dots ,5$,
leading to a total of 192 unitary matrices.'''
i understand that the example given forms a group under multiplication, i am looking to verify this using code so i need precise conditions that allows this set to become a group.
Thanks