Yes that's basically what abstract algebra is, it's defining how different actions work on sets of arbitrary elements. It get's pretty abstract pretty darn fast.
And I was talking to Eric about the zero stuff. He only knows Fields and VectorSpaces, he's a n00b
Oh, and an Algebra (in particular, a Lie Algebra) is something else entirely. Wikipedia is pretty good at explaining Abstract Algebra, once you know the terminology.
H*K is defined as {(h,k) for all h in H, k in K}
then you can define 'similar' functions, f1' f2' g1' and g2':
- f1': H*K -> H as f1'(h,k) = h
- f2': H*K -> K as f2'(h,k) = k
- g1': H -> H*K as g1'(h) = (h,e)
- g2': K -> H*K as g2'(k) = (1,k)
where 1 and e are Identity elements in H and K respectively
Then you can show that these 'similar' functions act the same as the first ones:
- f1' o g1' = Ih OR g1'(f1'(h,e)) = g1'(h) = (h,e)
- f2' o g2' = Ik OR g2'(f2'(1,k)) = g2'(k) = (1,k)
- f1' o g2' = 0' OR g2'(f1'(h,k)) = g2'(h) = (1,e)
- f2' o g1' = 0' OR g1'(f2'(h,k)) = g1'(k) = (1,e)
and
- (f1' o g1') + (f2' o g2') = i' OR g1'(f1'(h,k)) + g2'(f2'(h,k)) = (h,e) + (1,k) = (h,k)
Where 0' is the zero map and i' is the identity map over H*K.
I can't remember the correct words to use though, since it's been a year or so since I did this stuff. Basically these new 'similar' functions are enough to let you prove that G and H*K are similar, aka isomorphic. I'd suggest to not take anything I've said as rigorous proof, but I'm fairly sure that this is the way one should approach such a problem.