## Exhibit the set of all invertible diagonal matrices with diagonal entries equal as a direct product

Let , where is a field, be the group of upper triangular matrices all of whose diagonal entries are equal. Prove that , where is the group of nonzero multiples of the identity matrix and the group of strictly upper triangular matrices.

Recall that is normal in , where denotes the upper triangular matrices with only 1 on the diagonal. Then is normal in . Now if , we have for some nonzero field element . Then for all , we have , so that ; more specifically, is normal in .

We can see that is trivial, since every element in has only 1s on the main diagonal. Finally, if has the element on the main diagonal, then , and in fact . Thus ; by the recognition theorem, we have .

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By

nbloomf, on

June 22, 2010 at 11:00 am, under

AA:DF,

Incomplete. Tags:

center (group),

diagonal matrix,

direct product,

general linear group,

normal subgroup,

recognition theorem,

upper triangular matrix. 4 Comments

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## Comments

U is NOT SLn(F) intersect G.

True- it should be the strictly upper triangular elements in . Should be fixed now- thanks!

Where can I recall that is normal in ? Was it in the D&F?

Bah- I thought we showed that, but now I can’t find it. I’ll have to mark this incomplete and come back to it.

We did show previously that is normal in , that might play a role in the fix.