Degree homomorphism of a compact connected Lie group

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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.