Connection between A(N-1) and SU(N)

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According to this book I'm reading these groups are said to be equivalent and I am trying to better understand how this is so.

The generators of SU(N) are N^2-1 traceless complex anti-hermitian matrices. The group is over the real numbers and is N^2-1 dimensional.

On the other hand we have the group A(N-1), a subgroup of GL(N,C). The generators of A(N-1) are N^2-1 traceless real matrices but this over the complex numbers. If your require that the generating element be anti-hermitian this reduces the number of complex parameters from N^2-1 to (N^2-1)/2. I.e. we have N^2-1 real parameters and hence the groups are identical.

Is this the correct way to view this?
 
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Well I can see no response then I shall rephrase the question.
Is it true, and if so why is it true, that the groups A(N-1) and SU(N) have the same root system? i.e. Isomorphic algebras? What is the appropriate group morphism? etc.
 

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