Can someone tell me if the following proposition is true ?
let $G$ be group and $g_i\in G$ for $i=1\ldots n$ for any permutation $\sigma\in S_n$ then we have $g_1g_2\ldots g_ng^{-1}_{\sigma(1)}\ldots g^{-1}_{\sigma(n)} \in G'$
Can someone tell me if the following proposition is true ?
let $G$ be group and $g_i\in G$ for $i=1\ldots n$ for any permutation $\sigma\in S_n$ then we have $g_1g_2\ldots g_ng^{-1}_{\sigma(1)}\ldots g^{-1}_{\sigma(n)} \in G'$
Let $f:G\to G/G'$ be the canonical surjection. Let $n\geq1$, $g_1,\dots,g_n\in G$, and $\sigma\in S_n$. Since the quotient $G/G'$ is abelian, we have \begin{align} f(g_1\cdots g_ng_{\sigma(1)}^{-1}\cdots g_{\sigma(1)}^{-1}) &= f(g_1)\cdots f(g_n)f(g_{\sigma(1)}^{-1})\cdots f(g_{\sigma(1)}^{-1})) \\\\ &= f(g_1)f(g_{\sigma(1)}^{-1})\cdots f(g_n)f(g_{\sigma(n)}^{-1}) \\\\ &= 0 \end{align} so that $g_1\cdots g_ng_{\sigma(1)}^{-1}\cdots g_{\sigma(1)}^{-1}$ is in the kernel of $f$, namely, in $G'$.