group theory

# Contents

## Definition

### For abelian groups

###### Definition

For $A$, $B$ and $C$ abelian groups and $A×B$ the cartesian product group, a bilinear map

$f:A×B\to C$f : A \times B \to C

from $A$ and $B$ to $C$ is a function of the underlying sets (that is, a binary function from $A$ and $B$ to $C$) which is a linear map – that is a group homomorphism – in each argument separately.

###### Remark

In terms of elements this means that a bilinear map $f:A×B\to C$ is a function of sets that satisfies for all elements ${a}_{1},{a}_{2}\in A$ and ${b}_{1},{b}_{2}\in B$ the two relations

$f\left({a}_{1}+{a}_{2},{b}_{1}\right)=f\left({a}_{1},{b}_{1}\right)+f\left({a}_{2},{b}_{1}\right)$f(a_1 + a_2, b_1) = f(a_1,b_1) + f(a_2, b_1)

and

$f\left({a}_{1},{b}_{1}+{b}_{2}\right)=f\left({a}_{1},{b}_{1}\right)+f\left({a}_{1},{b}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$f(a_1, b_1 + b_2) = f(a_1, b_1) + f(a_1, b_2) \,.

Notice that this is not a group homomorphism out of the direct product group. The product group $A×B$ is the group whose elements are pairs $\left(a,b\right)$ with $a\in A$ and $b\in B$, and whose group operation is

$\left({a}_{1},{b}_{1}\right)+\left({a}_{2},{b}_{2}\right)=\left({a}_{1}+{a}_{2}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}{b}_{1}+{b}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2 \;,\; b_1 + b_2) \,.
$\varphi :A×B\to C$\phi : A \times B \to C

hence satisfies

$\varphi \left({a}_{1}+{a}_{2},{b}_{1}+{b}_{2}\right)=\varphi \left({a}_{1},{b}_{1}\right)+\varphi \left({a}_{2},{b}_{2}\right)$\phi( a_1+a_2, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(a_2, b_2)

and hence in particular

$\varphi \left({a}_{1}+{a}_{2},{b}_{1}\right)=\varphi \left({a}_{1},{b}_{1}\right)+\varphi \left({a}_{2},0\right)$\phi( a_1+a_2, b_1 ) = \phi(a_1,b_1) + \phi(a_2, 0)
$\varphi \left({a}_{1},{b}_{1}+{b}_{2}\right)=\varphi \left({a}_{1},{b}_{1}\right)+\varphi \left(0,{b}_{2}\right)$\phi( a_1, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(0, b_2)

which is (in general) different from the behaviour of a bilinear map.

The definition of tensor product of abelian groups is precisely such that the following is an equivalent definition of bilinear map:

###### Definition

For $A,B,C\in \mathrm{Ab}$ a function of sets $f:A×B\to C$ is a bilinear map from $A$ and $B$ to $C$ precisely if it factors through the tensor product of abelian groups $A\otimes B$ as

$f:A×B\to A\otimes B\to C\phantom{\rule{thinmathspace}{0ex}}.$f : A \times B \to A \otimes B \to C \,.
###### Remark

The analogous defintion for more than two arguments yields multilinear maps. There is a multicategory of abelian groups and multilinear maps between them; the bilinear maps are the binary morphisms, and the multilinear maps are the multimorphisms.

### For modules

More generally :

###### Definition

For $R$ a ring (or rig) and $A,B,C\in R$Mod being modules (say on the left, but on the right works similarly) over $R$, a bilinear map from $A$ and $B$ to $C$ is a function of the underlying sets

$f:A×B\to C$f : A \times B \to C

which is a bilinear map of the underlying abelian groups as in def. 1 and in addition such that for all $r\in R$ we have

$f\left(ra,b\right)=rf\left(a,b\right)$f(r a, b) = r f(a,b)

and

$f\left(a,rb\right)=ff\left(a,b\right)\phantom{\rule{thinmathspace}{0ex}}.$f(a, r b) = f f(a,b) \,.

As before, this is equivalent to $f$ factoring through the tensor product of modules

$f:A×B\to A{\otimes }_{R}B\to C\phantom{\rule{thinmathspace}{0ex}}.$f : A \times B \to A \otimes_R B \to C \,.

Multilinear maps are again a generalisation.

## Examples

• For $R=k$ a field, an $R$-module is a $k$-vector space and a $R$-bilinear map is a bilinear map out of two vector spaces.

## References

In the context of higher algebra/(∞,1)-category theory bilinear maps in an (∞,1)-category are discussed in section 4.3.4 of

Revised on February 11, 2013 21:18:12 by Urs Schreiber (89.204.138.151)