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 Incoming links: Fundamental group of a topological group

Eckmann-Hilton argument Created 2026-09-24 Updated 2026-11-05

Suppose two binary operations

*

and

\cdot

on one set share an identity element

e

and satisfy the interchange rule

(a * b) \cdot (c * d) = (a \cdot c) * (b \cdot d) .

Then the operations coincide, because

a \cdot b = (a * e) \cdot (e * b) = (a \cdot e) * (e \cdot b) = a * b .

They are also commutative:

a \cdot b = (e * a) \cdot (b * e) = (e \cdot b) * (a \cdot e) = b * a = b \cdot a .

These substitutions prove the Eckmann-Hilton argument without cancellation assumptions. In the fundamental group of a topological group, the two operations are loop concatenation and pointwise group multiplication. They share the constant loop as identity on homotopy classes, so the fundamental group is Abelian.

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