Find the explicit representation of the right and left cosets of the indicated subgroups H of GL(n). In each case, determine whether the left and right cosets corresponding to the same A ∈ GL(n) are equal.
(i) H = SO(n),
(ii) H = c(Id); c ∈ R \ {0},
(iii) H consists of n × n diagonal matrices with nonzero determinant.
I get the latter part of the question by checking the normality of each subgroup H. However, I'm not sure how to write "explicit representation" of the cosets. Is there more explicit way than simply expressing it as, for example: (i) Left coset = {AB; A ∈ GL(n), B ∈ SO(n)}?
(for instance)
(i) The coset of a matrix $A$ is the set of all matrices $B$ such that $BB^T=AA^T$ and $\det(B)=\det(A)$.
The first condition comes from $A^{-1}B$ being orthogonal, the second from it being special.
(ii) The coset of a matrix $A$ is the $1$-dimensional subspace spanned by $A$ in $\mathcal{M}_{n\times n}$, deprived from $0_{n\times n}$.
(iii) The coset of $A$ is the set of all matrices whose columns are individually non-zero multiples of the corresponding columns of $A$.