Group of matrix isomorphism

I have 2 groups:

• general linear $k \times k$ with $\cdot$
• top-triangle matrix $n \times n$ with 1 on main diagonal. Operation is $\cdot$ too

Is there isomorphism for any any non-trivial $n,k$ i.e $n \neq 2 \ or \ k \neq 1$ over $\mathbb{R}$ or $\mathbb{Q}$?

If no, how can I prove it?

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Did you intend $\mathbb{N}$ to be the set of integers? natural numbers? –  Bill Cook Dec 15 '11 at 13:20
@BillCook, I mixed it up with $\mathbb{Q}$, fixed now –  RiaD Dec 15 '11 at 13:21
Upper-triangular matrices form solvable groups, general linear groups are not solvable (for $k>1$). Thus they cannot be isomorphic. –  Bill Cook Dec 15 '11 at 13:22
@BillCook, Hmm, ok. Can you post as an answer? I'll accept it –  RiaD Dec 15 '11 at 13:27

Upper-triangular matrices form solvable groups, general linear groups are not solvable (for $k>1$). Thus they cannot be isomorphic.
Another reason is that if the field is of characteristics $0$ then all elements (except the identity matrix) in the set upper triangle matrix with 1 on the main diagonal do not have finite order. However, there are lots of matrix in $GL_k(F)$ has finite order. For instance, those have $-1$ or $1$ on the main diagonal and $0$ elsewhere.