/sci/ - Science & Math

Hi,

I've got a bit of trouble with understanding the bijectivity in isomorphisms.
Of course I know what a bijective function is, but in this case, I'm not sure if this applies the way I think it does.

Think of two groups <span class="math"> (G_1, \circ_1), (G_2, \circ_2)[/spoiler], where G is the group and <span class="math"> \circ [/spoiler] an operation.

A function f is a homomorphism, if for every element x, y, in <span class="math"> H_1 [/spoiler] it holds true that

<span class="math"> f(x \circ_1 y) = f(x) \circ_2 f(y) [/spoiler]

If f is also bijective, then it's an isomorphism between two groups.
But what does that actually mean? The isomorphism is a function that maps one algebraic structure to another, but does the bijectivity of the function apply to how the elements of the groups are being mapped to each other or to the actual groups?

I'd appreciate it if anyone could shed some light on that!