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- W4298274325 abstract "The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as well as $p$-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of $p$-regular classes or the number of $p$-singular classes of the group. In particular, if $G$ is a finite group with $O_p(G)=1$ then $|G/F(G)|_{p'}$ is bounded in terms of the number of $p$-regular classes of $G$. However, it is not possible to prove that there are finitely many groups with no nontrivial normal $p$-subgroup and $k$ $p$-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite)." @default.
- W4298274325 created "2022-10-01" @default.
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- W4298274325 date "2013-08-04" @default.
- W4298274325 modified "2023-09-23" @default.
- W4298274325 title "Variations of Landau's theorem for p-regular and p-singular conjugacy classes" @default.
- W4298274325 doi "https://doi.org/10.48550/arxiv.1308.0848" @default.
- W4298274325 hasPublicationYear "2013" @default.
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