Diagonal subgroup is self-centralizing in general linear group
This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
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(If has only two elements, the diagonal subgroup is trivial, and so clearly is not self-centralizing for ).
Given: A field with more than two elements, the group of invertible matrices, the subgroup of invertible diagonal matrices.
To prove: is self-centralizing in .
Proof: Suppose has the property that commutes with every element of . We want to show that .
Suppose not. Then, there exists a nonzero off-diagonal entry of , say the entry. Consider a diagonal matrix with different entries in the and places (for this, we need the field to have more than two elements). Then, and have different values in the position, hence . Hence, does not commute with every element of , leading to the required contradiction.