In Algebra, we just finished chapter 10 of Gallian. One of the big theorems in the chapter is the First Isomorphism Theorem – and since I’m a lazy typist I’ll call it FIT from now on. (Isomorphism theorems 2 and 3 are left as exercises.) Skipping over what a group is, what a homomorphism is, what the kernel of said homomorphism is, and what normal subgroups and factors groups are (see, FIT talks about all sorts of cool stuff!), FIT says:

Suppose is a homomorphism. Then ; moreover, the map by is an isomorphism.

One of the points of the proof, since is defined on the cosets of by specifying the image of a representative, is whether or not the map is even well defined for some In particular, will taking different representatives of the coset always have the same image? It turns out that the map is indeed well defined – it’s a rather straightforward argument – but this just brings up another question:

If a map is well defined, does it have the property of well-definededness?

Yay, English!

EDIT: Fixed my accidental interspersing of for

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