Why does the subgroup have to be able divide the group? For example why isn't the group:
$S= \{-4,-3,-1,0,1,2,3,4\}$
a subgroup of $G= \{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6 \}$? Aren't both additive, have a neutral number, each member has a reciprocal and every member in group $S$ is present in group $G$?
Why do subgroups have such a specific requirement of being be able to divide the group?