# nLab tensor product of groups

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Definition

###### Definition

Let $M,N$ be a pair of groups (not necessarily abelian) endowed with actions by group automorphisms on each other: $\rho$ an action of $M$ on $N$, and $\sigma$ of $N$ on $M$.

Then the tensor product $M \otimes N$ is the group generated by elements of the form $m\otimes n$ subject to the following relations:

• $(m m')\otimes n \,=\, (m m' m^{-1} \otimes \rho(m)(n))(m\otimes n),$

• $m\otimes (n n') \,=\, (m\otimes n)(\sigma(n)(m)\otimes n n' n^{-1} ),$

for $m,m'\in M$ and $n,n'\in N$.

This definition is due to Brown & Loday 1987, Section 2 (there in the context of the van Kampen theorem).

###### Remark

Def is to be understood as the generalization of the tensor product of abelian groups, since as one can verify, whenever $M,N$ act trivially on each other, then the above definition reduces to the ordinary tensor product of the abelianizations $(-)^{op}$ of the given groups:

$M\otimes N\cong M^{ab} \otimes_{\mathbb{Z}} N^{ab}.$

## References

The definition first appears in:

Last revised on January 26, 2024 at 05:03:06. See the history of this page for a list of all contributions to it.