The non-coprime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>k</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">GV</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> problem

Guralnick, Robert M.; Maróti, Attila

doi:10.1016/j.jalgebra.2005.02.001

Cited by 26 publications

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“…Then k(HV ) = r i=1 k(C H (v i )). We remark that Proposition 7.3 may be generalized to the situation when the order of H is divisible by p. See a result of Guralnick and Tiep [GT,Corollary 2.5…”

Section: Affine Groups With Few Conjugacy Classesmentioning

confidence: 93%

On almost p-rational characters of p ′ p^{\prime} -degree

2022

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Let p be a prime and let G be a finite group. A complex character of G is called almost p-rational if its values belong to a cyclotomic field ℚ ⁢ ( e 2 ⁢ π ⁢ i / n ) {{\mathbb{Q}}(e^{2\pi i/n})} for some n ∈ ℤ + {n\in{\mathbb{Z}}^{+}} not divisible by p 2 {p^{2}} . We prove that, in contrast to usual p-rational characters, there are “many” almost p-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost p-rational irreducible characters of G in terms of p. In fact, motivated by the McKay–Navarro conjecture, we obtain the same bound for the number of such characters of p ′ {p^{\prime}} -degree and prove that, in the minimal situation, the number of almost p-rational irreducible p ′ {p^{\prime}} -characters of G coincides with that of 𝐍 G ⁢ ( P ) {{\mathbf{N}}_{G}(P)} for P ∈ Syl p ⁡ ( G ) {P\in{\operatorname{Syl}}_{p}(G)} . Lastly, we propose a new way to detect the cyclicity of Sylow p-subgroups of a finite group G from its character table, using almost p-rational irreducible p ′ {p^{\prime}} -characters and the blockwise refinement of the McKay–Navarro conjecture.

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Section: Affine Groups With Few Conjugacy Classesmentioning

confidence: 93%