On the Existence of a Sporadic Composition Factor, Isomorphic to \(J_2\), \(McL\) or \(HS\) in Finite Groups with the Condition on the Prime Graph

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/e0260-9676-5868-v On the Existence of a Sporadic Composition Factor, Isomorphic to \(J_2\), \(McL\) or \(HS\) in Finite Groups with the Condition on the Prime Graph Zinovieva, M. R. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3. Abstract: The prime graph (or the Gruenberg–Kegel graph) of a finite group \(G\) is a simple graph \(GK(G)\) whose vertices are the prime divisors of the order of \(G\), and two distinct vertices \(p\) and \(q\) are adjacent in \(GK(G)\) if and only if \(G\) contains an element of order \(pq\). The prime graphs of non-abelian finite simple groups are known. One of the most popular fields of research in finite group theory is the study of finite groups by properties of their prime graphs. We study non-abelian composition factors of finite groups with the same prime graphs as non-abelian finite simple groups. In 2011, A. M. Staroletov studied groups having the same spectrum (the set of all orders of their elements) as a finite simple group and a sporadic composition factor. Generalizing this result, we consider whether the composition factor of a finite group with the same prime graph as the prime graph of a finite simple group can be isomorphic to a sporadic group. More exactly, we investigate whether it can be one of the sporadic groups \(J_2\), \(McL\) or \(HS\). Keywords: finite group, composition factor, group of Lie type, alternating group, sporadic group, prime graph (Gruenberg-Kegel graph) Language: Russian Download the full text Download JATS XML For citation: Zinovieva, M. R. On the Existence of a Sporadic Composition Factor, Isomorphic to \(J_2\), \(McL\) or \(HS\) in Finite Groups with the Condition on the Prime Graph, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp.74-90 (in Russian). DOI 10.46698/e0260-9676-5868-v + References 1. Kondrat'ev, A. S. Finite Groups with Given Properties of Their Prime Graphs, Algebra and Logic, 2016, vol. 55, no. 1, pp. 77-82. DOI: 10.1007/s10469-016-9378-5. 2. Staroletov, A. M. Sporadic Composition Factors of Finite Groups Isospectral to Simple Groups, Siberian Electronic Mathematical Reports, 2011, no. 8, pp. 268-272. 3. Vasil'ev, A. V. and Vdovin, E. P. An Adjacency Criterion for the Prime Graph of a Finite Simple Group, Algebra and Logic, 2005, vol. 44, no. 6, pp. 381-406. DOI: 10.1007/s10469-005-0037-5. 4. Zinovieva, M. R. 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On Recognition of Alternating Groups by Prime Graph, Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 994-1010. DOI: 10.17377/semi.2017.14.084. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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