Show that $U(8)$ is isomorphic to the group of matrices.

Question

I am trying to solve this question and I used (https://math.stackexchange.com/q/1677456)'s answer, but I don't fully understand two things:

1. Why does it matter that $a^{2}=b^{2}=c^{2}=1$?
2. Can $a, b, c$ be mapped to any matrix besides the identity? Does that mean this mapping is defined explicitly for each element?

Answer 1

1. $a^2 = b^2 = c^2 = 1$ just means that all nonidentity elements have order 2 in the group given that our group is $\{ a,b,c,1 \}$.
2. $a, b$ and $c$ can be mapped arbitrarily in this group provided they don't map to to the same element, or the identity. We cannot send group elements to group elements arbitrarily in general, it just worked out this way for this isomorphism. Also note that isomorphisms need not be unique, so should someone solve this problem, they may send elements to different elements than you did when you solved it.