# Degree homomorphism of a compact connected Lie group

## Definition

Let $G$ be a compact connected Lie group. Then the degree homomorphism from $G$ to $G$ is a homomorphism of multiplicative monoids:

$\mathbb{Z} \to \mathbb{Z}$

that sends an integer $d$ to the degree of the map $g \mapsto g^d$.

The degree homomorphism is a homomorphism of multiplicative monoids, because the degree of a composite mapping is the product of the degrees of the mappings. In particular, it sends 0 to 0 and 1 to 1.

The degree homomorphism can be used to compute the degree of any map from $G$ to $G$ defined by a word. This is because if $w(x)$ is a word involving an indeterminate $x$, then all the letters of $w$ other than $x$ or $x^{-1}$, can be homotoped to the identity element.