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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.

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Your example seems more topological than group-theoretical to me (both are fundamental to Lie theory, of course). – anon Mar 16 '12 at 10:54
In particular, the maximal tori in the example don't characterize diagonal matrices. – Ralph Mar 16 '12 at 10:59
son, that group ain't normal. best to put thoughts of such things out of yer mind... – David Wheeler Mar 16 '12 at 17:59
up vote 7 down vote accepted

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.

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