Exercise 27, § 2

A group $G$ is said to have the property $P(n)$ (n an integer > 0) if, for each system $(s_1, \ldots, s_n)$ of $n$ elements of $G$, there is a permutation $\sigma \in S_n$, other than the identity permutation (but depending on $s_1, \ldots, s_n$) such that $s_{\sigma(1)} \cdots s_{\sigma(n)} = s_1 \cdots s_n$.

a) Show that the non-commutative group $S_3$ satisfies $P(4)$.

b) Let $G$ be a connected Hausdorff topological group. Show that if $G$ satisfies $P(n)$ for some integer $n > 1$, then $G$ is commutative. [Show that $G$ satisfies $P(n - 1)$: given $n - 1$ elements $s_1, \ldots, s_{n-1}$ of $G$, reduce to the case where there is a neighbourhood $U$ of $e$ such that, for each $x \in U$, we have $s_1 \cdots s_{n-1} x = s_1 \cdots s_i x s_{i+1} \cdots s_{n-1}$ for some index $i = i(x)$; deduce that the centralizer in $G$ of one of the elements $s_{j+1} \cdots s_{n-1}$ ($0 \leq j \leq n - 2$) is both open and closed and therefore equal to $G$. Hence the result if $j \geq 1$; if $j = 0$, show that