Isotopic groups are isomorphic

Statement
The following are equivalent for groups $$G$$ and $$H$$:


 * 1) They are isotopic as magmas, i.e., there exists an fact about::isotopy of magmas between $$G$$ and $$H$$.
 * 2) They are fact about::isomorphic magmas, i.e., there exists an fact about::isomorphism of magmas between $$G$$ and $$H$$.
 * 3) They are fact about::isomorphic groups, i.e., there exists an fact about::isomorphism of groups between $$G$$ and $$H$$.

Related facts

 * Weaker than::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.

Facts used

 * 1) uses::Homotopy of groups arises from a homomorphism
 * 2) uses::Equivalence of definitions of isomorphism of groups

Proof
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).