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## Main Question or Discussion Point

Can someone explain to me how to find all the cosets of a set like H={A in GL(n) | det(A) = 1} in GL(n) (set of invertible n x n matrices)?

It's obvious how to find all the cosets for something simple like 3Z (set of all multiples of 3) in Z, we just find elements in Z, but not in 3Z that partitions Z, namely choosing 0, 1, 2 as those elements in Z gives the cosets 0+3Z, 1+3Z, 2+3Z, which partitions Z.

It's also obvious in the case where the group is finite. But how will I find it for H in GL(n) above? I'm not sure how to choose elements g in GL(n) but not in H, such that all the cosets of the form gH will partition GL(n). Any suggestions?

It's obvious how to find all the cosets for something simple like 3Z (set of all multiples of 3) in Z, we just find elements in Z, but not in 3Z that partitions Z, namely choosing 0, 1, 2 as those elements in Z gives the cosets 0+3Z, 1+3Z, 2+3Z, which partitions Z.

It's also obvious in the case where the group is finite. But how will I find it for H in GL(n) above? I'm not sure how to choose elements g in GL(n) but not in H, such that all the cosets of the form gH will partition GL(n). Any suggestions?

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