Universal quadratic functor

Definition
The universal quadratic functor (sometimes called Whitehead's universal quadratic functor) is a functor from abelian groups to abelian groups defined as follow. For an abelian group $$G$$ it outputs a group $$\Gamma(G)$$ given as the quotient of a free group on all the symbols $$\gamma(x), x \in G$$ by the following types of relations:


 * $$\gamma(0)= 0$$ (this condition is redundant)
 * $$\gamma(-x) = \gamma(x) \ \forall \ x \in G$$
 * $$\gamma(x) + \gamma(y) + \gamma(z) + \gamma(x + y + z) = \gamma(y + z) + \gamma(x + y) + \gamma(x + z) \ \forall x,y,z \in G$$.

Note that the above set of relations is equivalent to the following pair of assumptions:


 * The mapping $$x \mapsto \gamma(x)$$ is homogeneous of degree two: $$\gamma(nx) = n^2 \gamma(x)$$ for all $$n \in \mathbb{Z}$$
 * The mapping $$(x,y) \mapsto \gamma(x + y) - \gamma(x) - \gamma(y)$$ is a bihomomorphism, i.e., it is additive in each coordinate.

Facts

 * The exponent of $$\gamma(x)$$ in $$\Gamma(G)$$ divides twice the exponent of $$x$$ in $$G$$. This follows from noting that the bilinear form $$b(x,y) = \gamma(x + y) - \gamma(x) - \gamma(y)$$ also satisfies $$b(x,-x) = -2\gamma(x)$$ so $$b(x,x) = 2\gamma(x)$$, and the exponent of $$b(x,x)$$ divides the exponent of $$x$$ due to biadditivity.
 * Formula for universal quadratic functor of direct product