## 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.

### Equivalence of definitions

Further information: equivalence of definitions of universal quadratic functor

## 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

## Particular cases

Starting group $G$ Universal quadratic functor $\Gamma(G)$ Comments
finite cyclic group of odd order $n$, i.e., $\mathbb{Z}_n$ $\mathbb{Z}_n$ Intuitively, this is saying that if $x$ is defined mod $n$ for $n$ odd, then $x^2$ is defined mod $n$ but we cannot do better in general
finite cyclic group of even order $n$, i.e., $\mathbb{Z}_n$ $\mathbb{Z}_{2n}$ Intuitively, this is saying that if $x$ is defined mod $n$ for $n$ even, then $x^2$ is defined mod $2n$ but we cannot do better in general
group of integers $\mathbb{Z}$ $\mathbb{Z}$ we can think of the generator for $\Gamma(G)$ as the squaring map in the ring of integers