A question of order of the product of two elements

[rulin]

Let $X = \{1,2,\cdots, p\}$ with $p$ prime, and $Y = \{1,2,\cdots,t,i_1,i_2,\cdots,i_{p - t}\}$ with $1\leq t\leq p - 1$ and $i_j\not\in X$ for $1\leq j\leq p - t$. Supose that $a$ and $b$ are both $p$-rotation in $X$ and $Y$, respectively. Whether or not $p$ don't divide the order of $ab$?

 

[confinement]

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