Problem 5.1.3 ($xy = yx$ for all $x \in G_I$ and all $y \in G_K$ if $I \cap K = \{1\}$)

Proof. Let $x=(g_1,g_2,\dots ,g_n)\in G_I$ and $y=(h_1,h_2,\dots ,h_n)\in G_K$, so that $g_i=1$ if $i\notin J$ and $h_i=1$ if $i\notin K$.

Since $I\cap K=\varnothing$, $([[1,n]]-I)\cup ([[1,n]]-K)=[[1,n]]$, therefore, for all $i\in [[1,n]]$,

Hence

for all $i\in [[1,n]]$, and so

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