Euler characteristic of compact connected nontrivial Lie group is zero

Statement
Suppose $$G$$ is a nontrivial compact connected fact about::Lie group. Then, $$G$$ is a fact about::space with zero Euler characteristic, i.e., the fact about::Euler characteristic of $$G$$ is $$0$$.

Similar facts

 * Euler characteristic of odd-dimensional compact connected orientable manifold is zero

Facts used

 * 1) uses::Lefschetz fixed-point theorem
 * 2) uses::Lie group implies polyhedron: Any Lie group can be given a simplicial complex structure, and is hence a polyhedron.
 * 3) Definition of Euler characteristic as the Lefschetz number of the identity map from a space to itself.
 * 4) uses::Lefschetz number is homotopy-invariant: If $$f_1,f_2:X \to Y$$ are homotopic maps, then the Lefschetz numbers of $$f_1$$ and $$f_2$$ are equal.

Proof
Given: A compact connected nontrivial Lie group $$G$$.

To prove: The Euler characteristic of $$G$$ is zero.

Proof: We denote by $$e$$ the identity element of $$G$$.