Example 1

Let X be an arcwise connected, unlaceable, separated topological space, and let $a$ be a point of X. Let $n$ be an integer $\geqslant 2$ and let Y be the quotient of the space $X^n$ by the action of the group $\mathfrak{S}_n$ operating by permutation of the factors; let $f: X^n\rightarrow Y$ denote the canonical mapping; let $g: X\rightarrow Y$ denote the mapping $x\mapsto f(x, a, . . . , a)$. It follows from the proposition that, for every $i$, the homomorphism $\pi_1(g, a)$ of $\pi_1(X, a)$ into $\pi_1(Y, g(a))$ is surjective and that its kernel is the derived group of $\pi_1(X, a)$. In particular, the group $\pi_1(Y, g(a))$ is abelian.