Abstract
L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group Ak must have degree at least \(k^2(\frac{1}{2}-o(1))\), and has asked how sharp this lower bound is. We prove that Babai’s bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree \(\frac{3}{2}k(k-1)\). We also reprove Babai’s quadratic lower bound with the constant 1/2 improved to 1 (by completely different methods).
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Dedicated to our friend and colleague Alex Lubotzky
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Guralnick, R.M., Liebeck, M.W. Permutation representations of nonsplit extensions involving alternating groups. Isr. J. Math. 229, 181–191 (2019). https://doi.org/10.1007/s11856-018-1794-x
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DOI: https://doi.org/10.1007/s11856-018-1794-x