Finite groups and degrees of irreducible monomial characters

Pang, Linna; Lu, Jiakuan

doi:10.1142/s0219498816500730

Cited by 12 publications

(6 citation statements)

References 4 publications

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“…Interestingly, there seems to be a parallel between results that can be proved using only monolithic characters and results about solvable groups that can be proved using only monomial characters. For example, Pang and Lu proved in [15] that if G is solvable and p does not divide χ(1) for all monomial χ ∈ Irr(G), then P is normal in G.…”

Section: Introductionmentioning

confidence: 99%

Monolithic Brauer Characters

Chen

Lewis²

2019

Bull. Aust. Math. Soc.

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Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

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Section: Introductionmentioning

confidence: 99%

Monolithic Brauer Characters

Chen

Lewis²

2019

Bull. Aust. Math. Soc.

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“…This has been verified for all solvable groups G with |cd(G)| ≤ 5 ([9, Main Theorem]) and for all finite groups of odd order ([1, Theorem 2.4]). However, much less is known about how the set mcd(G) influences G. Linna Pang and Jiakuan Lu recently made progress in this direction (see [11] and [12]), and we shall use their results heavily.…”

Section: Introductionmentioning

confidence: 99%

On super-monomial characters and groups having two irreducible monomial character degrees

Færgeman

2019

Preprint

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A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd M -group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd M -group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.

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“…Throughout this paper, all groups are finite. In [4], Pang and Lu show that the properties of solvable groups coming from the degrees of the irreducible characters can be determined using only the degrees of the monomial irreducible characters. In other words, for solvable groups, the monomial irreducible characters are plentiful enough to be used in place of the irreducible characters.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Pang and Lu prove in [4,Theorem 1.3] that if G is solvable and p is a prime, then p does not divide χ(1) for every monomial character χ ∈ Irr(G) if and only if G has a normal Sylow p-subgroup. Hence, Pang and Lu are able to generalise the normal Sylow subgroup portion of Itô's theorem [1,Corollary 12.34].…”

Section: Introductionmentioning

confidence: 99%