Subgroups of GL2 Consider the subgroups of for some field . Then, the group quotient can be found by taking arbitrary elements in , multiplying them together, and figuring out what structure this gives. SoNote the determinant condition on implies , hence is a subgroup (which are the matrices where ). For fixed , the linear equation implies , which is an arbitrary element in since . Since we can take any matrix in and multiply it by the matrixwith , we can get a diagonal matrix in . This shows the quotient group .
Subgroups of GL2 Consider the subgroups of for some field . Then, the group quotient can be found by taking arbitrary elements in , multiplying them together, and figuring out what structure .this gives. SoNote the determinant condition on implies , hence is a subgroup (which are the matrices where ). For fixed , the linear equation implies , which is an arbitrary element in since  Since we can take any matrix in and multiply it by the matrixwith , we can get a diagonal matrix in . This shows the quotient group .