Tensor power of a group

Definition

Suppose $G$ is a group and $n$ is a positive integer. The $n^{th}$ tensor power of $G$, which we may denote as $\bigotimes ^{n}G$, is defined inductively as follows. Note that at each stage, the $n^{th}$ tensor power has the structure of a crossed module over $G$, and it is important to store this structure in order to induct rather than simply storing the group as an abstract group.

Base case

The first tensor power of $G$ is $G$ itself. The crossed module structure is as follows: the action by automorphisms is the action by conjugation, and the homomorphism is the identity map.

Inductive step

Suppose we have defined $\bigotimes ^{n}G$ as a crossed module over $G$. Then, define $\bigotimes ^{n+1}G$ as follows:

• First, note that since crossed module defines a compatible pair of actions, we have a natural choice of a compatible pair of actions between $\bigotimes ^{n}G$ and $G$.
• As an abstract group, define $\bigotimes ^{n+1}G$ as the tensor product $\bigotimes ^{n}G\otimes G$ for this compatible pair of actions.
• Now use the fact that tensor product of groups has a crossed module structure with respect to each group to give $\bigotimes ^{n+1}G$ the structure of a crossed module over $G$.