Regular orbits in powers of permutation representations

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Let (Q,G) be a faithful permutation representation of a finite group G. Suppose that the G-set Q has t distinct non-zero marks. In a permutation representation analogue of a theorem of Brauer on linear representations, it is shown that the direct power (Q,G) t of (Q,G) contains a regular orbit. As a corollary, the probability that a random element of Q r lies in a regular orbit of (Q,G) r is shown to tend to 1 exponentially fast as r tends to \(\infin\). Further, knowledge of the rate of convergence is equivalent to knowledge of the second largest value of the character of the linear permutation representation.