Isotopic groups are isomorphic
The following are equivalent for groups and :
- They are isotopic as magmas, i.e., there exists an Isotopy of magmas (?) between and .
- They are Isomorphic magmas (?), i.e., there exists an Isomorphism of magmas (?) between and .
- They are Isomorphic groups (?), i.e., there exists an Isomorphism of groups (?) between and .
- Group implies G-loop: This states that if a group and a loop are isotopic, then they are isomorphic, and in particular, the algebra loop itself is a group.
The equivalence of (1) and (2) follows from fact (1). Specifically, if a homotopy of groups is an isotopy, the homomorphism giving rise to it must also be bijective and hence must be an isomorphism. The equivalence of (2) and (3) follows from fact (2).