Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 7 / c / Solution

Use the stable finite-dimensional description. Define addition by block sum, $[u]+[v]=[\mathrm{diag}(u,v)]$. Stabilizations and homotopies preserve block sums, so this is well-defined. Associativity is immediate after identifying the three blocks; the identity class is zero.

Swapping two blocks is conjugation by a permutation unitary. Every finite-dimensional unitary is joined to the identity, since the matrix spectral theorem writes it as $e^{iA}$ with $A$ self-adjoint. Conjugation along that path gives a homotopy swapping the blocks, proving commutativity.

The inverse of $[u]$ is $[u^{-1}]$. To verify it rather than assuming a group structure, first stabilize to equal-sized blocks and putThe invertible homotopystarts at $\mathrm{diag}(u,u^{-1})$ and ends at the identity. Thus $\boxed{K^1(X)\text{ is an abelian group}.}$ The constructions are continuous in $x$ and remain valid for maps into $G{L}_N$; using polar factors also makes them homotopies in the unitary model.