Proving prime conditions for Sylow p-subgroups in finite groups, indicating deeper analytic implications.
Fix an integer m>2. We prove that if there exists a finite group with mp+1 Sylow p-subgroups, where p is large enough, then mp+1 is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer’s Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.
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Urroz et al. (2026) studied this question.
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