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The block graph of a finite group

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Abstract

In this paper we first define the block graph of a finite group G, whose vertices are the prime divisors of ∣G∣ and there is an edge between two vertices p ≠ q if and only if the principal p- and q-blocks of G have a nontrivial common complex irreducible character of G. Then we determine the block graphs of finite simple groups, which turn out to be complete except those of J1 and J4. Finally, in terms of block graphs with no triangle containing a prime p, we obtain a criterion for the p-solvability of a finite group which in particular leads to an equivalent condition for the solvability of a finite group.

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Acknowledgements

Most of the work was done during the visit of the second author at the TU Kaiserslautern from July 2016 to July 2017. The authors are deeply grateful to Professor Jiping Zhang for pointing out to us Theorem 1.4, to Professor Wolfgang Willems for his suggestion about the problems in this paper, and to Professor Gunter Malle for his invaluable advice and support. Also, we deeply thank the referee for his/her important comments on a previous version of this paper.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Julian Brough & Alessandro Paolini

  2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

    Yanjun Liu

Authors
  1. Julian Brough
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  2. Yanjun Liu
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  3. Alessandro Paolini
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Corresponding author

Correspondence to Yanjun Liu.

Additional information

The first and the third authors gratefully acknowledge financial support by the ERC Advanced Grant 291512, and the third author also gratefully acknowledges financial support by the SFB-TRR195. The second author deeply thanks financial support by China Scholarship Council (201608360074), the National Natural Science Foundation of China (11661042 & 11761034), and the Natural Science Foundation of Jiangxi Province (20192ACB21008).

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Brough, J., Liu, Y. & Paolini, A. The block graph of a finite group. Isr. J. Math. 244, 293–317 (2021). https://doi.org/10.1007/s11856-021-2192-3

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  • DOI: https://doi.org/10.1007/s11856-021-2192-3

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