A question of order of the product of two elements

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