Regular orbits in powers of permutation representations |
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Authors: | JDH Smith |
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Institution: | Department of Mathematics, Iowa State University, Ames, IA 50011, USA, US
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Abstract: | 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 Qr lies in a regular orbit of (Q,G)r is shown to tend to 1 exponentially fast as r tends to \infin\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. |
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