403 research outputs found
Nonvanishing elements for Brauer characters
No full textLet G be a finite group and p a prime. We say that a p-regular element g of G is p-nonvanishing if no irreducible p-Brauer character of G takes the value 0 on g. The main result of this paper shows that if G is solvable and g is a p-regular element which is p-nonvanishing, then g lies in a normal subgroup of G whose p-length and p'-length are both at most 2 (with possible exceptions for p\leq 7), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for p>7) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113]
Groups whose prime graph on class sizes has a cut vertex
No full text[EN] Let G be a finite group, and let Delta(G) be the prime graph built on the set of conjugacy class sizes of G: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of G, two vertices p and q being adjacent if and only if pq divides some conjugacy class size of G. In the present paper, we classify the finite groups G for which Delta(G) has a cut vertex.The research of the first and second authors is partially supported by the Italian PRIN 2015TW9LSR_006 "Group Theory and Applications" and by INdAM-GNSAGA.
The research of the third author is supported by the Spanish Ministerio de Ciencia e Innovacion PID2019-103854GB-I00 partly with FEDER funds.
The fourth author acknowledges the support of the Spanish Ministerio de Ciencia, Innovacion y Universidades proyecto PGC2018-096872-B-I00, the grant ACIF/2016/170 from Generalitat Valenciana, and the prize Borses Ferran Sunyer i Balaguer 2019.Dolfi, S.; Pacifici, E.; Sanus, L.; Sotomayor, V. (2021). Groups whose prime graph on class sizes has a cut vertex. Israel Journal of Mathematics. 244(2):775-805. https://doi.org/10.1007/s11856-021-2193-2S7758052442D. Bubboloni, S. Dolfi, M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, Journal of Pure and Applied Algebra 213 (2009), 1722–1734.C. Casolo and S. Dolfi, The diameter of a conjugacy class graph of finite groups, Bulletin of the London Mathematical Society 28 (1996), 141–148.C. Casolo and S. Dolfi, Products of primes in conjugacy class sizes and irreducible character degrees, Israel Journal of Mathematics 174 (2009), 403–418.C. Casolo, S. Dolfi, E. Pacifici and L. Sanus, Groups whose prime graph on conjugacy class sizes has few complete vertices, Journal of Algebra 364 (2012), 1–12.C. Casolo, S. Dolfi, E. Pacifici and L. Sanus, Incomplete vertices in the prime graph on conjugacy class sizes of finite groups, Journal of Algebra 376 (2013), 46–57.S. Dolfi, Arithmetical conditions on the length of the conjugacy classes of a finite group, Journal of Algebra 174 (1995), 753–771.S. Dolfi, On independent sets in the class graph of a finite group, Journal of Algebra 303 (2006), 216–224.S. Dolfi, E. Pacifici, L. Sanus and V. Sotomayor, The prime graph on class sizes of a finite group has a bipartite complement, Journal of Algebra 542 (2020), 35–42.I. M. Isaacs, Coprime group actions fixing all nonlinear irreducible characters, Canadian Journal of Mathematics 41 (1989), 68–82.I. M. Isaacs, Finite Group Theory, Graduate Studies in mathematics, Vol. 92, American Mathematical Society, Providence, RI, 2008.M. L. Lewis and Q. Meng, Solvable groups whose prime divisor character degree graphs are 1-connected, Monatshefte für Mathematik 190 (2019), 541–548.O. Manz and T. R. Wolf, Representations of Solvable Groups, London Mathematical Society Lecture Note Series, Vol. 185, Cambridge University Press, Cambridge, 1993.J. M. Riedl, Character degrees, class sizes and normal subgroups of a certain class of p-groups, Journal of Algebra 218 (1999), 190–215
Groups whose character degree graph has diameter three
No full textLet G be a finite group, and let Δ(G) denote the prime graph built on the
set of degrees of the irreducible complex characters of G. It is well known
that, whenever Δ(G) is connected, the diameter of Δ(G) is at most 3. In
the present paper, we provide a description of the finite solvable groups
for which the diameter of this graph attains the upper bound. This also
enables us to confirm a couple of conjectures proposed by M. L. Lewis
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