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Degrees of irreducible representations of direct products of nonabelian simple groups

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Abstract

For a finite group G, let the character degree set of G, denoted by \(\mathrm{cd}(G)\), be the set of the degrees of all irreducible complex representations of G. In the present paper, the structure of finite groups whose character degree sets coincide with the character degree sets of direct product of nonabelian simple groups will be studied. For a finite group S and a positive integer n, let \(S^{n}\) be the direct product of n copies of S. We prove that if G is a finite group with \(\mathrm{cd}(G)=\mathrm{cd}(H)\), where

$$\begin{aligned}&H\in \{\mathrm{PSL}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q-1), \mathrm{PSU}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q+1), \\&\quad M_{11}^{n} (1\leqslant n\leqslant 2), M_{23}^{n}, J_{1}^{n}, J_{2}^{n}, J_{3}^{n}, J_{4}^{n}\}, \end{aligned}$$

then G is a quasi perfect group. This extends the first step of Huppert’s Conjecture to the direct product of simple groups. This conjecture states that the nonabelian simple groups are uniquely determined up to an abelian direct factor by the set of character degrees.

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Correspondence to Kamal Aziziheris.

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The author would like to thank the University of Tabriz for supporting of this work.

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Aziziheris, K. Degrees of irreducible representations of direct products of nonabelian simple groups. RACSAM 114, 68 (2020). https://doi.org/10.1007/s13398-020-00799-2

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Keywords

  • Simple group
  • Character degree
  • Quasi perfect group
  • Direct product of simple groups

Mathematics Subject Classification

  • Primary 20C15
  • Secondary 20D05