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# {#section-9.unnumbered} If $C=\{1,b\}$ and $D=\{c,d\}$ are double cosets, then we have $$D=\{k_1b+c,l_1b+d\}=\{l_1c,l_1d\}=\{l_1,1\}=D’=\{c,d\} \quad \textrm{ for } \quad c,d\in\mathbb{Z}_n^*,$$ $$D=\{kb,l_1b\}=\{l_1k_1,k_1l_2\}=\{l_2,1\}=D’,$$ $$C=\{kb+c,l_1b+d\}=\{c,d\}=C’, \quad \textrm{ for } \quad c,d\in\mathbb{Z}_n^*,$$ $$C=\{1+kb,l_1b\}=\{b,1\}=C’, \quad \textrm{ for } \quad b\in\mathbb{Z}_n^*,$$ $$C=\{k_1b+c,1+l_1b\}=\{b,1\}=C’,$$ $$C=\{c,d\}=C’, \quad \textrm{ for } \quad c,d\in\mathbb{Z}_n^*.$$ In the proof of The