Prime divisors of irreducible character degrees

Tong-Viet, Hung P.

doi:10.1216/rmj-2015-45-5-1645

Cited by 4 publications

(9 citation statements)

References 18 publications

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“…As the last preliminary result we recall the statement of Theorem A in [19], that proves the strengthened ρ-σ conjecture for almost-simple groups.…”

Section: Preliminariesmentioning

confidence: 82%

“…One might wonder whether the factor 3 is the right one for non-solvable groups, or one should instead keep the factor 2 and add a suitable constant for getting a tighter bound. In a recent paper ( [19]), H. Tong-Viet studies the so-called strengthened ρ-σ conjecture proposed by Manz and Wolf in [16], that is |ρ(G)| ≤ 2σ(G) + 1 for every finite group G. The strengthened ρ-σ conjecture is verified in [19] for all finite almost-simple groups and also for the groups G such that σ(G) ≤ 2.…”

Section: Introductionmentioning

confidence: 94%

“…To the end of showing σ(S) ≥ 3, by Lemma 2.4 in [19] we only have to rule out the case S ∼ = PSL 2 (q) (where q is a p-power with p > 3) and |π(q ± 1)| ≤ 2; but in this situation the degree of (say) χ 1 , as defined in the second paragraph of this proof, is divisible only by 2 and 3, so π(χ 1 ) − π(C) is empty, against (i). As for σ(C) ≤ 4, this follows at once by the fact that, as observed, we have σ(C) < σ(G); the remaining conclusions also follow easily, taking into account that (by (iv)) C contains a simple normal subgroup of G.…”

Section: The ρ-σ Conjecture For Groups With Trivial Fitting Subgroupmentioning

confidence: 99%

“…Next, we sketch the proof of a result concerning almost-simple groups that can be essentially deduced from the proofs of Lemma 2.2 and Theorem 2.3 in [19] (we refer the reader to that paper for the full details).…”

Section: Preliminariesmentioning

confidence: 99%

“…Next, assume that at least one among the primitive prime divisors ℓ 1 , ℓ 2 indicated in Table 1 (which summarizes [19,Table 1] and [18, Lemma 2.3]) does not exist; then, it can be seen that S is isomorphic to a group of the kind PSL 2 (q), PSL 3 (q), PSU 3 (q) or PSp 4 (q) for some specific values of q = p f , or S belongs to a finite set of groups (see [19,List C,page 3]). Also, π(G) turns out to coincide with π(S) in this situation.…”

Section: Preliminariesmentioning

confidence: 99%

See 4 more Smart Citations

On Huppert's Rho-Sigma Conjecture

Akhlaghi¹,

Dolfi²,

Pacifici³

2021

Preprint

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For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)| ≤ 3σ(G) always holds, whereas |ρ(G)| ≤ 2σ(G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a "strengthened" form of the conjecture in the general case, asking whether |ρ(G)| ≤ 2σ(G) + 1 is true for every finite group G. In this paper we study the strengthened ρ-σ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided σ(G) ≤ 5, but it is false in general if σ(G) ≥ 6. Instead, we establish that |ρ(G)| ≤ 3σ(G) − 4 holds for every finite group with a trivial Fitting subgroup and with σ(G) ≥ 6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have |ρ(G)| ≤ 3σ(G) whenever G belongs to one particular class including all the finite solvable groups.

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“…As the last preliminary result we recall the statement of Theorem A in [19], that proves the strengthened ρ-σ conjecture for almost-simple groups.…”

Section: Preliminariesmentioning

confidence: 82%