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.