nLab tensor product of groups

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:

• Ronnie Brown, Jean-Louis Loday. Van Kampen theorems for diagrams of spaces. Topology 26 3 (1987) 311-335 [doi:10.1016/0040-9383(87)90004-8]