# Universal quadratic functor

From Groupprops

## 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 it outputs a group given as the quotient of a free group on all the symbols by the following types of relations:

- (this condition is redundant)
- .

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

- The mapping is homogeneous of degree two: for all
- The mapping 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 in divides twice the exponent of in . This follows from noting that the bilinear form also satisfies so , and the exponent of divides the exponent of due to biadditivity.
- Formula for universal quadratic functor of direct product

## Particular cases

Starting group | Universal quadratic functor | Comments |
---|---|---|

finite cyclic group of odd order , i.e., | Intuitively, this is saying that if is defined mod for odd, then is defined mod but we cannot do better in general | |

finite cyclic group of even order , i.e., | Intuitively, this is saying that if is defined mod for even, then is defined mod but we cannot do better in general | |

group of integers | we can think of the generator for as the squaring map in the ring of integers |