# Homotopy of groups arises from a homomorphism

## Contents

## Statement

Suppose and are groups, and are maps such that is a Homotopy of magmas (?) from to . In other words, for all , we have:

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 .

## Related facts

### Corollaries

## Proof

**Given**: Groups , a homotopy

**To prove**: There is a homomorphism such that:

**Proof**: Consider the defining equation of a homotopy of magmas:

Setting gives:

Setting gives:

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.