Homotopy of groups arises from a homomorphism
Then, there is a Homomorphism of groups (?) such that:
where is the identity element of .
In particular, since left and right multiplication maps are bijective, all the maps have the same patterns of fibers as the homomorphism , namely, cosets of the kernel of .
Given: Groups , a homotopy
To prove: There is a homomorphism such that:
Proof: Consider the defining equation of a homotopy of magmas:
Combining the above, we get:
This simplifies to:
Let be this common value. Then and . Also, from the above relation, we obtain that .
It remains to show that the map is a homomorphism. For this, consider the original defining relation:
Writing everything in terms of , we obtain:
Canceling and we obtain:
Thus, is a homomorphism of groups.