Finite groups contain large centralizers
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2021
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Springer
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Palacín, D. Finite groups contain large centralizers. Israel Journal of Mathematics 2021, 244(2), 621–624. doi:10.1007/s11856-021-2183-4.
Abstract
Every finite non-abelian group of order n has a non-central element whose centralizer has order exceeding n^{1/3}. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.