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Abstract Letk(B0) andl(B0) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB0of a finite groupG. We prove that, ifk(B0)−l(B0) = 1, thenl(B0) ≥p− 1 or elsep= 11 andl(B0) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B0) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ . These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ , wherekp(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups.
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null (Ed.)Abstract We show that every finite group of order divisible by 2 or q , where q is a prime number, admits a $$\{2, q\}'$$ { 2 , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /q})$$ Q ( e 2 π i / q ) . We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /p})$$ Q ( e 2 π i / p ) or $${\mathbb{Q}}(e^{ 2 \pi i/q})$$ Q ( e 2 π i / q ) , for any pair of primes p and q . Along the way, we completely describe simple alternating groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with rational values.more » « less
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Abstract The classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p , then G has a normal Sylow p -subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.more » « less
