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 .