Are G & H Isomorphic? Prove Answer | Group of 2x2 Matrices

[symbol0]

Let G be the group of 2x2 invertible upper triangular matrices and H be the group of 2x2 invertible lower triangular matrices (both groups under multiplication). Are G and H isomorphic? Prove answer.

First I thought they were isomorphic, but couldn't find an isomorphism, so now I believe they are not isomorphic, but can't pinpoint exactly why.
Any suggestions?

 

[Tinyboss]

Transposition is a good first guess for an isomorphism, since the transpose of an upper triangular matrix is lower triangular, and vice-versa. But transposition isn't a homomorphism, since (AB)^T=B^TA^T (the order gets reversed). Kind of like how (AB)^{-1}=B^{-1}A^{-1}. Can you combine them?

 

[symbol0]

Tinyboss, Why do you say it is a good first guess. As you mention, the order gets reversed.
I don't think they are isomorphic. I know that if there was an isomorphism f, then f(-A)= -f(A) for all matrices A in G. Also, f would map the subgroup D of diagonal matrices to itself.

 

[Simon_Tyler]

Read the last sentence in tinyboss's reply...

 

[symbol0]

Ohh, of course.
Thank you guys.