Find the group G generated under matrix multiplication by the matrices
$A=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}.$
Determine its proper subgroups, and verify for each of them that its cosets exhaust G.

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Accepted Answer

Before we can draw up a group multiplication table to search for subgroups, we must determine the multiple products of A and B with themselves and with each other:
[latex]{A}^{2}=\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}=I[/latex].
Since B = iA, it follows that [latex]{B}^{2} = −I[/latex], that AB = iI = BA, and that [latex]{B}^{3} = −B[/latex]. In brief, A is of order 2, B is of order 4, and A and B commute. The eight distinct elements of the group are therefore: [latex]I,A,B,{B}^{2},{B}^{3}, AB,A{B}^{2}[/latex] and [latex]A{B}^{3}[/latex].
The group multiplication table is
By inspection, the subgroups and their cosets are as follows:
{I,A} : {I,A}, {B,AB}, {[latex]{B}^{2},A{B}^{2}[/latex]}, {[latex]{B}^{3},A{B}^{3}[/latex]};
{[latex]I,{B}^{2}[/latex]} : {[latex]I,{B}^{2}[/latex]}, {[latex]A,A{B}^{2}[/latex]}, {[latex]B,{B}^{3}[/latex]}, {[latex]AB,A{B}^{3}[/latex]};
{[latex]I,A{B}^{2}[/latex]} : {[latex]I,A{B}^{2}[/latex]}, {[latex]A,{B}^{2}[/latex]}, {[latex]B,A{B}^{3}[/latex]}, {[latex]{B}^{3},AB[/latex]};
{[latex]I,B,{B}^{2},{B}^{3}[/latex]} : {[latex]I,B,{B}^{2},{B}^{3}[/latex]}, {[latex]A,AB,A{B}^{2},A{B}^{3}[/latex]};
{[latex]I,AB,{B}^{2},A{B}^{3}[/latex]} : {[latex]I,AB,{B}^{2},A{B}^{3}[/latex]}, {[latex]A,B,A{B}^{2},{B}^{3}[/latex]}.
As expected, in each case the cosets exhaust the group, with each element in one and only one coset.

Question : Find the group G generated under matrix multiplication by th...

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