/sci/ - Science & Math

[OP !!yzPI6QwWJCg No.5385956]

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!

[Anonymous No.5385961]

>>5385956
You can think of an isomorphism as a 'relabeling' of the elements in the group. the bijectivity part just ensures your two groups are actually the same size.

[OP !!yzPI6QwWJCg No.5385966]

>>5385961
..because in a bijective function, you can be sure that from every image you can "find back" to a single preimage, meaning there are no open ends?

[Anonymous No.5385972]

>>5385966
Sure, you can think about it like that.

[OP !!yzPI6QwWJCg No.5385975]

>>5385972
Alright, thanks!

[Anonymous No.5386021]

>>5385961
>>5385966
This is about it. Two groups are "the same" essentially (from an algebraic perspective) if they have the same "amount" (quotes used to take into account infinite sets) of elements and those elements interact with each other the same way in both groups. That last part is the operation-function definition you gave in the post.