{"title":"特征度图具有割顶点的不可解群。三、","authors":"Silvio Dolfi, Emanuele Pacifici, Lucia Sanus","doi":"10.1007/s10231-023-01328-9","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a finite group. Denoting by <span>\\(\\textrm{cd}(G)\\)</span> the set of the degrees of the irreducible complex characters of <i>G</i>, we consider the <i>character degree graph</i> of <i>G</i>: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in <span>\\(\\textrm{cd}(G)\\)</span>, and two distinct vertices <i>p</i>, <i>q</i> are adjacent if and only if <i>pq</i> divides some number in <span>\\(\\textrm{cd}(G)\\)</span>. This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a <i>cut-vertex</i>, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor <i>S</i>, and that <i>S</i> is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for <i>S</i> were treated, except the case <span>\\(S\\cong \\textrm{PSL}_{2}(2^a)\\)</span> for some integer <span>\\(a\\ge 2\\)</span>; the remaining case is addressed in the present paper.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-023-01328-9.pdf","citationCount":"1","resultStr":"{\"title\":\"Non-solvable groups whose character degree graph has a cut-vertex. III\",\"authors\":\"Silvio Dolfi, Emanuele Pacifici, Lucia Sanus\",\"doi\":\"10.1007/s10231-023-01328-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a finite group. Denoting by <span>\\\\(\\\\textrm{cd}(G)\\\\)</span> the set of the degrees of the irreducible complex characters of <i>G</i>, we consider the <i>character degree graph</i> of <i>G</i>: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in <span>\\\\(\\\\textrm{cd}(G)\\\\)</span>, and two distinct vertices <i>p</i>, <i>q</i> are adjacent if and only if <i>pq</i> divides some number in <span>\\\\(\\\\textrm{cd}(G)\\\\)</span>. This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a <i>cut-vertex</i>, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor <i>S</i>, and that <i>S</i> is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for <i>S</i> were treated, except the case <span>\\\\(S\\\\cong \\\\textrm{PSL}_{2}(2^a)\\\\)</span> for some integer <span>\\\\(a\\\\ge 2\\\\)</span>; the remaining case is addressed in the present paper.</p></div>\",\"PeriodicalId\":8265,\"journal\":{\"name\":\"Annali di Matematica Pura ed Applicata\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-04-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10231-023-01328-9.pdf\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annali di Matematica Pura ed Applicata\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10231-023-01328-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-023-01328-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Non-solvable groups whose character degree graph has a cut-vertex. III
Let G be a finite group. Denoting by \(\textrm{cd}(G)\) the set of the degrees of the irreducible complex characters of G, we consider the character degree graph of G: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in \(\textrm{cd}(G)\), and two distinct vertices p, q are adjacent if and only if pq divides some number in \(\textrm{cd}(G)\). This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor S, and that S is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for S were treated, except the case \(S\cong \textrm{PSL}_{2}(2^a)\) for some integer \(a\ge 2\); the remaining case is addressed in the present paper.
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