The number of set-orbits of a solvable permutation group
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Abstract A permutation group 𝐺 acting on a set Ω induces a permutation action on the power set P ( Ω ) \mathscr{P}(\Omega) (the set of all subsets of Ω). Let 𝐺 be a finite permutation group of degree 𝑛, and let s ( G ) s(G) denote the number of orbits of 𝐺 on P ( Ω ) \mathscr{P}(\Omega) . In this paper, we give the explicit lower bound of log 2 s ( G ) / log 2 | G | \log_{2}s(G)/{\log_{2}\lvert G\rvert} over all solvable groups 𝐺. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ p^{\prime} -degree irreducible representations of a solvable group.
2012 ◽
Vol 56
(1)
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pp. 303-336
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2012 ◽
Vol 56
(2)
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pp. 371-386
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2016 ◽
Vol 104
(1)
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pp. 37-43
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2001 ◽
Vol 71
(3)
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pp. 349-352
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2012 ◽
Vol 87
(3)
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pp. 406-424
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1990 ◽
Vol 21
(2)
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pp. 309-310
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