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A characterization of the prime graphs of solvable groups. (English) Zbl 1331.20029

Let \(G\) be a finite group and \(\pi(G)\) be the set of prime divisors of \(|G|\). The prime graph \(\Gamma_G\) of \(G\), is the graph with vertex set \(\pi(G)\) and edges \(\{p,q\}\in E(\Gamma_G)\) if and only if there exists an element of order \(pq\) in \(G\).
The main theorem in this paper is Theorem 2: An unlabeled graph \(\mathcal G\) is isomorphic to the prime graph of some finite solvable group if and only if its complement \(\overline{\mathcal G}\) is \(3\)-colorable and triangle-free.
An essential tool for the proof of the results contained in this paper is the so-called Lucido’s Three Primes Lemma [M. S. Lucido, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 5, No. 1, 131-148 (2002; Zbl 1097.20022)] which asserts that if \(G\) is a finite solvable group and \(p,q,r\in\pi(G)\), then \(G\) contains an element of order the product of two of these three primes.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C15 Coloring of graphs and hypergraphs
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Citations:

Zbl 1097.20022
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References:

[1] Chartrand, Gary; Zhang, P., Chromatic Graph Theory, Discrete Math. Appl. (2008), Taylor & Francis
[2] Chvátal, Vašek, The minimality of the Mycielski graph, (Graphs and Combinatorics. Graphs and Combinatorics, Lecture Notes in Math., vol. 406 (1974), Springer: Springer Berlin, Heidelberg), 243-246
[3] Greenwood, Robert E.; Gleason, Andrew Mattei, Combinatorial relations and chromatic graphs, Canad. J. Math., 7, 1, 7 (1955) · Zbl 0064.17901
[4] Hall, P., A note on soluble groups, J. Lond. Math. Soc., s1-3, 2, 98-105 (1928) · JFM 54.0145.01
[5] Higman, Graham, Groups and rings having automorphisms without non-trivial fixed elements, J. Lond. Math. Soc., s1-32, 3, 321-334 (1957) · Zbl 0079.03203
[6] Huppert, B., Character Theory of Finite Groups, de Gruyter Exp. Math. (1998), Walter de Gruyter · Zbl 0932.20007
[7] Keller, Thomas Michael, Solvable groups with a small number of prime divisors in the element orders, J. Algebra, 170, 2, 625-648 (1994) · Zbl 0816.20022
[8] Keller, Thomas Michael, A linear bound for \(\rho(n)\), J. Algebra, 178, 2, 643-652 (1995) · Zbl 0859.20014
[9] Kondratév, A. S., Prime graph components of finite simple groups, Sb. Math., 180, 6, 787-797 (1969) · Zbl 0691.20013
[10] Kondratév, A. S.; Khramtsov, I. V., On finite tetraprimary groups, Proc. Steklov Inst. Math., 279, 1, 43-61 (2012) · Zbl 1303.20025
[11] Lovász, L., Three short proofs in graph theory, J. Combin. Theory, 19, 269-271 (1975) · Zbl 0322.05142
[12] Lucido, Maria Silvia, The diameter of a prime graph of a finite group, J. Group Theory, 2, 2, 157-172 (1999) · Zbl 0921.20020
[13] Lucido, Maria Silvia, Groups in which the prime graph is a tree, Bol. Unione Mat. Ital. Sez. B, 5, 1, 131-148 (2002) · Zbl 1097.20022
[14] Moghaddamfar, Ali Reza; Zokayi, A. R., Recognizing finite groups through order and degree pattern, Algebra Colloq., 15, 3, 449-456 (2008) · Zbl 1157.20015
[15] Vasilév, A. V., On connection between the structure of a finite group and the properties of its prime graph, Sib. Math. J., 46, 3, 396-404 (2005)
[16] Vasilév, A. V.; Vdovin, E. P., An adjacency criterion for the prime graph of a finite simple group, Algebra Logic, 44, 6, 381-406 (2005)
[17] Williams, J. S., Prime graph components of finite groups, J. Algebra, 69, 487-513 (1981) · Zbl 0471.20013
[18] Zavarnitsine, A. V., Finite groups with a five-component prime graph, Sib. Math. J., 54, 1, 40-46 (2013) · Zbl 1275.20009
[19] Zinovéva, M. R.; Mazurov, V. D., On finite groups with disconnected prime graph, Proc. Steklov Inst. Math., 283, 1, 139-145 (2013)
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