Fundamental group of a topological group

For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.

If two unital operations on one set share their identity and satisfy $(a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d)$, then the operations coincide and are commutative.

For $n\ge 1$, pointwise multiplication of based cube maps into a topological group induces the ordinary addition in its homotopy group ${\pi }_n(G,e)$. Concatenation and pointwise multiplication share an identity and satisfy the interchange rule, so the Eckmann-Hilton argument identifies the two operations. In particular, the $m$-th power map induces multiplication by $m$ on every homotopy group, for all integers $m$.