Proposition 3.5.0.1. Let $X$ be a connected Kan complex, let $x \in X$ be a vertex, and let $n$ be a positive integer. Suppose that the homotopy groups $\pi _{m}(X,x)$ vanish for every positive integer $m \neq n$. Then $X$ is homotopy equivalent to an Eilenberg-MacLane space $\mathrm{K}(G,n)$ for some group $G$ (which is abelian if $n \geq 2$).
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