Two's complement arithmetic proof

Question

Can anyone prove that groups $G:=({0,1,2,3,4,5,6,7},+)$ and $H:=({0,1,2,3,-4,-3,-2,-1},+)$ are isomorphic with the bijection which is two's complement and + meaning modulo 8 addition? I mean, the fact that two's complement is a bijection is obvious but it's not obvious that $2c(a+b)=2c(a)+2c(b)$ for any a and b from $G$ or $H$. What I want is basically proof of validity of two's complement arithmetics. And I'm new to group theory (only wikipedia basics). TY.

Answer 1

Ok, both groups are isomorphic to $Z_8$, and $2c$ just makes the correct bijection between numbers representing corresponding congruence classes. This is not a rigid proof but that pretty much clears out the situation.