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Статті в журналах з теми "CHARACTER THEORY, FINITE GROUPS"

Автор: Grafiati

Опубліковано: 10 березня 2023

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1

Boyer, Robert. "Character theory of infinite wreath products." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1365–79. http://dx.doi.org/10.1155/ijmms.2005.1365.

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The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0-invariant of the groupC∗-algebra is also determined.

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2

Isaacs, I. M. "The π-character theory of solvable groups". Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, № 1 (серпень 1994): 81–102. http://dx.doi.org/10.1017/s1446788700036077.

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AbstractThere is a deeper structure to the ordinary character theory of finite solvable groups than might at first be apparent. Mauch of this structure, which has no analog for general finite gruops, becomes visible onyl when the character of solvable groups are viewes from the persepective of a particular set π of prime numbers. This purely expository paper discusses the foundations of this πtheory and a few of its applications. Included are the definitions and essential properties of Gajendragadkar's π-special characters and their connections with the irreducible πpartial characters and their associated Fong characters. Included among the consequences of the theory discussed here are applications to questions about the field generated by the values of a character, about extensions of characters of subgroups and about M-groups.

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3

Isaacs, I. M. "Book Review: Character theory of finite groups." Bulletin of the American Mathematical Society 36, no. 04 (July 22, 1999): 489–93. http://dx.doi.org/10.1090/s0273-0979-99-00789-2.

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4

Ikeda, Kazuoki. "Character Values of Finite Groups." Algebra Colloquium 7, no. 3 (August 2000): 329–33. http://dx.doi.org/10.1007/s10011-000-0329-1.

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5

Chetard, Béatrice I. "Graded character rings of finite groups." Journal of Algebra 549 (May 2020): 291–318. http://dx.doi.org/10.1016/j.jalgebra.2019.11.041.

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6

Schmid, Peter. "Finite groups satisfying character degree congruences." Journal of Group Theory 21, no. 6 (November 1, 2018): 1073–94. http://dx.doi.org/10.1515/jgth-2018-0019.

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Abstract Let G be a finite group, p a prime and {c\in\{0,1,\ldots,p-1\}} . Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p. If here {c=0} , then G has a normal p-complement by a well known theorem of Thompson. We prove that in the cases where {c\neq 0} the group G is solvable with a normal abelian Sylow p-subgroup. If {p\neq 3} then this is true provided these character degrees are congruent to c or to {-c} modulo p.

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7

Xiong, Huan. "Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles." Algebra Colloquium 23, no. 01 (January 6, 2016): 15–22. http://dx.doi.org/10.1142/s1005386716000031.

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Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud' in their book “Characters of Finite Groups”, we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character codegree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.

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8

Madanha, Sesuai Yash. "Zeros of primitive characters of finite groups." Journal of Group Theory 23, no. 2 (March 1, 2020): 193–216. http://dx.doi.org/10.1515/jgth-2019-2051.

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AbstractWe classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside’s classical theorem on zeros of characters.

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9

Qian, Guohua. "Finite groups with consecutive nonlinear character degrees." Journal of Algebra 285, no. 1 (March 2005): 372–82. http://dx.doi.org/10.1016/j.jalgebra.2004.11.021.

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10

Chillag, David, and Marcel Herzog. "Finite groups with almost distinct character degrees." Journal of Algebra 319, no. 2 (January 2008): 716–29. http://dx.doi.org/10.1016/j.jalgebra.2005.07.039.

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11

Cheng, Chuangxun. "A character theory for projective representations of finite groups." Linear Algebra and its Applications 469 (March 2015): 230–42. http://dx.doi.org/10.1016/j.laa.2014.11.027.

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12

Song, Xueling, and Yanjun Liu. "The Steinberg Character of Finite Classical Groups." Algebra Colloquium 20, no. 01 (January 16, 2013): 163–68. http://dx.doi.org/10.1142/s100538671300014x.

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Анотація:

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol).

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13

Malle, Gunter, and Alexandre Zalesski. "Steinberg-like characters for finite simple groups." Journal of Group Theory 23, no. 1 (January 1, 2020): 25–78. http://dx.doi.org/10.1515/jgth-2019-0024.

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AbstractLet G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called {\mathrm{Syl}_{p}}-regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension {\lvert S\rvert}, where F is an algebraically closed field of characteristic p.

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14

Wu, Yi-Tao, and Pu Zhang. "Finite solvable groups whose character graphs are trees." Journal of Algebra 308, no. 2 (February 2007): 536–44. http://dx.doi.org/10.1016/j.jalgebra.2006.09.009.

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15

Thiem, Nathaniel, and C. Ryan Vinroot. "Values of character sums for finite unitary groups." Journal of Algebra 320, no. 3 (August 2008): 1150–73. http://dx.doi.org/10.1016/j.jalgebra.2008.03.022.

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16

Isaacs, I. M. "Character kernels and degree ratios in finite groups." Journal of Algebra 322, no. 6 (September 2009): 2220–34. http://dx.doi.org/10.1016/j.jalgebra.2009.01.033.

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17

Qian, Guohua. "Nonsolvable groups with few primitive character degrees." Journal of Group Theory 21, no. 2 (March 1, 2018): 295–318. http://dx.doi.org/10.1515/jgth-2017-0037.

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18

NENCIU, ADRIANA. "BRAUER PAIRS OF VZ-GROUPS." Journal of Algebra and Its Applications 07, no. 05 (October 2008): 663–70. http://dx.doi.org/10.1142/s0219498808003065.

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Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. A group is called a VZ-group if all its nonlinear irreducible characters vanish off the center. In this paper we give necessary and sufficient conditions for two non-isomorphic VZ-groups to form a Brauer pair.

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19

Wolf, Thomas R. "Character Correspondences and π-Special Characters in π-Separable Groups". Canadian Journal of Mathematics 39, № 4 (1 серпня 1987): 920–37. http://dx.doi.org/10.4153/cjm-1987-046-1.

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Анотація:

Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups.

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20

Humphreys, J. F. "Character tables for the primitive finite unitary reflection groups." Communications in Algebra 22, no. 14 (January 1994): 5777–802. http://dx.doi.org/10.1080/00927879408825163.

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21

Li, Tianze, and Guohua Qian. "Finite Groups Whose Irreducible Character Degrees Constitute Two Chains." Communications in Algebra 41, no. 6 (May 21, 2013): 2100–2108. http://dx.doi.org/10.1080/00927872.2011.653465.

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22

Zhang, Jiping. "A note on character degrees of finite solvable groups." Communications in Algebra 28, no. 9 (January 2000): 4249–58. http://dx.doi.org/10.1080/00927870008827087.

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23

Li, Tianze, Yanjun Liu, and Xueling Song. "Finite nonsolvable groups whose character graphs have no triangles." Journal of Algebra 323, no. 8 (April 2010): 2290–300. http://dx.doi.org/10.1016/j.jalgebra.2010.01.019.

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24

Deshpande, Tanmay. "Shintani descent for algebraic groups and almost characters of unipotent groups." Compositio Mathematica 152, no. 8 (June 1, 2016): 1697–724. http://dx.doi.org/10.1112/s0010437x16007429.

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Анотація:

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

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25

Michler, G. O., та J. B. Olsson. "Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2". Canadian Journal of Mathematics 43, № 4 (1 серпня 1991): 792–813. http://dx.doi.org/10.4153/cjm-1991-045-6.

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In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

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26

Evans, Anthony B. "On Elementary Abelian Cartesian Groups." Canadian Mathematical Bulletin 34, no. 1 (March 1, 1991): 58–59. http://dx.doi.org/10.4153/cmb-1991-009-3.

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AbstractJ. Hayden [2] proved that, if a finite abelian group is a Cartesian group satisfying a certain "homogeneity condition", then it must be an elementary abelian group. His proof required the character theory of finite abelian groups. In this note we present a shorter, elementary proof of his result.

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27

Meng, Qingyun, and Jiwen Zeng. "Finite Groups Whose Character Degree Graphs Are Empty Graphs." Algebra Colloquium 20, no. 01 (January 16, 2013): 75–80. http://dx.doi.org/10.1142/s1005386713000060.

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28

López, Antonio Vera. "Conjugacy classes in finite groups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 105, no. 1 (1987): 259–64. http://dx.doi.org/10.1017/s0308210500022083.

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SynopsisIn this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

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29

Dabbaghian, Vahid, and John D. Dixon. "Computing characters of groups with central subgroups." LMS Journal of Computation and Mathematics 16 (October 2013): 398–406. http://dx.doi.org/10.1112/s1461157013000211.

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AbstractThe so-called Burnside–Dixon–Schneider (BDS) method, currently used as the default method of computing character tables in GAP for groups which are not solvable, is often inefficient in dealing with groups with large centres. If $G$ is a finite group with centre $Z$ and $\lambda $ a linear character of $Z$, then we describe a method of computing the set $\mathrm{Irr} (G, \lambda )$ of irreducible characters $\chi $ of $G$ whose restriction ${\chi }_{Z} $ is a multiple of $\lambda $. This modification of the BDS method involves only $\vert \mathrm{Irr} (G, \lambda )\vert $ conjugacy classes of $G$ and so is relatively fast. A generalization of the method can be applied to computation of small sets of characters of groups with a solvable normal subgroup.Supplementary materials are available with this article.

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30

LEWIS, MARK L. "LANDAU’S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS." Journal of the Australian Mathematical Society 104, no. 1 (December 23, 2016): 37–43. http://dx.doi.org/10.1017/s144678871600063x.

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When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

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31

Chen, Zhe. "Generic character sheaves on reductive groups over a finite ring." Journal of Pure and Applied Algebra 225, no. 3 (March 2021): 106521. http://dx.doi.org/10.1016/j.jpaa.2020.106521.

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32

Liu, Yanjun. "Finite groups with only one p -singular Brauer character degree." Journal of Pure and Applied Algebra 220, no. 9 (September 2016): 3182–206. http://dx.doi.org/10.1016/j.jpaa.2016.02.010.

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33

Levy, Dan. "The Average Sylow Multiplicity Character and Solvability of Finite Groups." Communications in Algebra 38, no. 2 (February 12, 2010): 632–44. http://dx.doi.org/10.1080/00927870902828694.

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34

Riese, Udo. "A Subnormality Criterion in Finite Groups Related to Character Degrees." Journal of Algebra 201, no. 2 (March 1998): 357–62. http://dx.doi.org/10.1006/jabr.1997.7259.

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35

Liang, Dengfeng, Guohua Qian, and Wujie Shi. "Finite groups whose all irreducible character degrees are Hall-numbers." Journal of Algebra 307, no. 2 (January 2007): 695–703. http://dx.doi.org/10.1016/j.jalgebra.2006.10.010.

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36

FUMA, MICHITAKU, and YASUSHI NINOMIYA. "FINITE GROUPS WITH MULTIPLICITY-FREE PERMUTATION CHARACTERS." Journal of Algebra and Its Applications 04, no. 02 (April 2005): 187–94. http://dx.doi.org/10.1142/s021949880500106x.

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Анотація:

Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End ℂ[G]((ℂH)G), where ℂH is the trivial ℂ[H]-module and (ℂH)G = ℂH⊗ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂH)G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).

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37

VINROOT, C. RYAN. "CHARACTER DEGREE SUMS AND REAL REPRESENTATIONS OF FINITE CLASSICAL GROUPS OF ODD CHARACTERISTIC." Journal of Algebra and Its Applications 09, no. 04 (August 2010): 633–58. http://dx.doi.org/10.1142/s0219498810004166.

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Let 𝔽q be a finite field with q elements, where q is the power of an odd prime, and let GSp (2n, 𝔽q) and GO ±(2n, 𝔽q) denote the symplectic and orthogonal groups of similitudes over 𝔽q, respectively. We prove that every real-valued irreducible character of GSp (2n, 𝔽q) or GO ±(2n, 𝔽q) is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if G is a classical connected group defined over 𝔽q with connected center, with dimension d and rank r, then the sum of the degrees of the irreducible characters of G(𝔽q) is bounded above by (q + 1)(d+r)/2. Finally, we show that if G is any connected reductive group defined over 𝔽q, for any q, the sum of the degrees of the irreducible characters of G(𝔽q) is bounded below by q(d-r)/2(q - 1)r. We conjecture that this sum can always be bounded above by q(d-r)/2(q + 1)r.

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38

Feng, Zhicheng, Conghui Li, Yanjun Liu, Gunter Malle, and Jiping Zhang. "Robinson’s conjecture for classical groups." Journal of Group Theory 22, no. 4 (July 1, 2019): 555–78. http://dx.doi.org/10.1515/jgth-2018-0177.

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AbstractRobinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

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39

Zhang, Jiping. "Finite Solvable Groups Whose Character Degree Graphs Are Not Complete." Algebra Colloquium 13, no. 04 (December 2006): 541–52. http://dx.doi.org/10.1142/s1005386706000496.

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Анотація:

In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert. Suppose that G is a finite solvable group and p is a prime number dividing the degree of some irreducible character of G. If there is another such prime number q such that pq does not divide the degree of any irreducible character of G, then both p-length ℓp(G) and q-length ℓq(G) of G are at most two, and ℓp(G)+ ℓq(G)=4 if and only if pq=6 with QG/Zφ(QG)≅ 32:GL(2,3), where QG is generated by all Sylow 2-subgroups of G and Zφ(G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.

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40

Cameron, Peter J., and Masao Kiyota. "Sharp characters of finite groups." Journal of Algebra 115, no. 1 (May 1988): 125–43. http://dx.doi.org/10.1016/0021-8693(88)90285-2.

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41

Qian, Guohua, та Yuanyang Zhou. "On 𝑝-parts of character codegrees". Journal of Group Theory 24, № 5 (31 березня 2021): 1005–18. http://dx.doi.org/10.1515/jgth-2020-0090.

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Анотація:

Abstract Let 𝐺 be a finite group and 𝑝 a prime. We define the codegree of χ ∈ Irr ⁢ ( G ) \chi\in\mathrm{Irr}(G) by cod ( χ ) = | G : ker χ | / χ ( 1 ) \operatorname{cod}(\chi)=\lvert G:\ker\chi\rvert/\chi(1) and define c p ( G ) = max { log p ( cod ( χ ) ) p ∣ χ ∈ Irr ( G ) } c_{p}(G)=\max\{\log_{p}(\operatorname{cod}(\chi))_{p}\mid\chi\in\mathrm{Irr}(G)\} . In this paper, we show that | G / O p ′ ⁢ p ⁢ ( G ) | p ≤ p c p ⁢ ( G ) \lvert G/O_{p^{\prime}p}(G)\rvert_{p}\leq p^{c_{p}(G)} for all finite groups 𝐺 and characterize the finite groups 𝐺 with c p ⁢ ( G ) ≤ 1 c_{p}(G)\leq 1 .

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42

QIAN, GUOHUA, and YANMING WANG. "ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS." Journal of Algebra and Its Applications 13, no. 02 (October 10, 2013): 1350100. http://dx.doi.org/10.1142/s0219498813501004.

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Анотація:

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

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43

Dabbaghian-Abdoly, Vahid. "Characters of some finite groups of Lie type with a restriction containing a linear character once." Journal of Algebra 309, no. 2 (March 2007): 543–58. http://dx.doi.org/10.1016/j.jalgebra.2006.10.021.

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44

Liu, Xiaolei. "On Lengths of Conjugacy Classes and Character Degrees in Finite Groups." Communications in Algebra 34, no. 4 (May 2006): 1443–49. http://dx.doi.org/10.1080/00927870500454976.

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45

Chang, Huimin, and Ping Jin. "Irreducible products of characters." Journal of Algebra and Its Applications 19, no. 04 (April 17, 2019): 2050069. http://dx.doi.org/10.1142/s0219498820500693.

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We introduce the notion of Fitting characters for arbitrary finite groups, and prove that under some conditions the product of these characters is irreducible and the unique factorization of this form also holds. Moreover, we show that any nonlinear quasi-primitive character of solvable groups can be uniquely factored (up to multiplication by linear characters) as the product of certain Fitting characters on some extension groups.

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46

Camina, Rachel D., Ainhoa Iñiguez, and Anitha Thillaisundaram. "Word problems for finite nilpotent groups." Archiv der Mathematik 115, no. 6 (July 17, 2020): 599–609. http://dx.doi.org/10.1007/s00013-020-01504-w.

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AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

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47

Feng, Zhicheng, Conghui Li, Yanjun Liu, Gunter Malle, and Jiping Zhang. "Robinson’s conjecture on heights of characters." Compositio Mathematica 155, no. 6 (May 20, 2019): 1098–117. http://dx.doi.org/10.1112/s0010437x19007267.

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Geoffrey Robinson conjectured in 1996 that the $p$-part of character degrees in a $p$-block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.

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48

Arad, Zvi, David Chillag, and Marcel Herzog. "Powers of characters of finite groups." Journal of Algebra 103, no. 1 (October 1986): 241–55. http://dx.doi.org/10.1016/0021-8693(86)90183-3.

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49

Holmes, Randall R. "Projective characters of finite Chevalley groups." Journal of Algebra 124, no. 1 (July 1989): 158–82. http://dx.doi.org/10.1016/0021-8693(89)90157-9.

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50

Inglis, Nicholas F. J. "Linear Characters of Finite Linear Groups." Journal of Algebra 214, no. 2 (April 1999): 545–52. http://dx.doi.org/10.1006/jabr.1997.7348.

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