Problem: Determine whether the given map

is an isomorphism of the first binary structure with the second.

with

where

is the determinant of matrix A.

My attempt:

I know that I need to complete four steps to prove isomorphism.

1) Define the function that gives isomorphism S to S'.

2) Show that

is one-to-one.

3) " " is onto S'.

4) Show that

for all

.

For step one,

det(x) for all

is in

, so

.

For step two,

If

, det(x) = det(y). This is false, since determinants are not unique. det(x) can equal det(y) while x is not equal to y. Therefore, this map is not isometric since it is not one-to-one.

Then I stop there since it's not one-to-one, there's no need to prove the rest. But am I right? And if so, how would I prove that it is or isn't onto?