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<p>Let <i>G</i> be a finite group and Irr(<i>G</i>) the set of all irreducible complex characters of <i>G</i>. Let cd(<i>G</i>) be the set of all irreducible complex character degrees of <i>G</i> and denote by <i>ϱ</i>(<i>G</i>) the set of all primes which divide a character degree of <i>G</i>. The character-prime graph Γ(<i>G</i>) associated to <i>G</i> is a simple undirected graph whose vertex set is <i>ϱ</i>(<i>G</i>) and there is an edge between two distinct primes <i>p</i> and <i>q</i> if and only if the <i>pq</i> divides a character degree of <i>G</i>. We show that the finite non-Abelian simple group <i>U</i><sub>3</sub>(7), <i>M</i><sub>11</sub>, <i>L</i><sub>2</sub>(16), <i>L</i><sub>2</sub>(25), <i>L</i><sub>2</sub>(81), <i>U</i><sub>3</sub>(8), <i>U</i><sub>3</sub>(9), <i>Sz</i>(8), <i>Sz</i>(32) and <i>L</i><sub>2</sub>(<i>p</i>) are uniquely determined by their degree-patterns and orders.</p>

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A new characterization of non-Abelian simple groups by their degree-patterns and orders

  • 期刊论文

  • 发表日期：

    2026年1月19日

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  • Shijie Tao,
  • Qianqian Wang,
  • Changguo Shao

摘要

Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G and denote by ϱ(G) the set of all primes which divide a character degree of G. The character-prime graph Γ(G) associated to G is a simple undirected graph whose vertex set is ϱ(G) and there is an edge between two distinct primes p and q if and only if the pq divides a character degree of G. We show that the finite non-Abelian simple group U3(7), M11, L2(16), L2(25), L2(81), U3(8), U3(9), Sz(8), Sz(32) and L2(p) are uniquely determined by their degree-patterns and orders.