Let $k$ be a field. Is there a group-theoretical characterization of the subgroup $D_n$ of diagonal matrices in $GL_n(k)$ ?
For example, if $k = \mathbb{C}\;$ then $D_n$ is a maximal torus, but, of course, there are many of them.
Let $k$ be a field. Is there a group-theoretical characterization of the subgroup $D_n$ of diagonal matrices in $GL_n(k)$ ?
For example, if $k = \mathbb{C}\;$ then $D_n$ is a maximal torus, but, of course, there are many of them.
There cannot be a group-theoretical characterization of the diagonal matrices, since every similarity transform is an automorphism of $GL_n(k)$, and similarity transforms generally don't leave the subgroup of diagonal matrices invariant.