One corollary of description of finite groups without elements of order 6. (Russian. English summary) Zbl 07896800
Otto Kegel and Karl Gruenberg introduced long time an interesting graph, which was useful to detect the structural properties of finite groups, looking at the order of their elements.
More precisely, if \(G\) is a finite group and the set of all prime divisors of the order of \(G\) is \(\pi(G)\), the “Gruenberg-Kegel graph” (also known in the literature with the name of “prime graph”) is the graph with vertices given by the elements of \(\pi(G)\) and two vertices \(p\) and \(q\) in \(\pi(G)\) are adjacent if and only if \(G\) contains an element of order \(pq\).
Some connected components of this graph were described very well by Maria Silvia Lucido in [M. S. Lucido, Rend. Semin. Mat. Univ. Padova 102, 1–22 (1999; Zbl 0941.20008)] and in some of her successive contributions. Essentially the role of the connected component of this graph containing the prime 2 is peculiar, but about the other components there isn’t a clear and uniform description of what happens in general (this is one of the “Problems” which are mentioned often in recent years and also in the paper under review by the authors).
It should be noted that the prime graph has been the subject of several generalizations, as illustrated by [F. G. Russo, “Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group”, Adv. Pure Math. 2, 391–396 (2012)], or by [A. F. Vasil’ev et al., Sib. Math. J. 63, No. 5, 849–861 (2022; Zbl 1516.20055); translation from Sib. Mat. Zh. 63, No. 5, 1010–1026 (2022)] or even in the context of infinite profinite groups by [W. Herfort et al., Topology Appl. 263, 26–43 (2019; Zbl 1442.22006)] and in many other contributions in group theory.
In the paper under review, the authors offer a description of the component of the prime graph focusing on the role of the prime 3 and considering the behaviour of the nonsolvable finite groups. It is interesting to note that among all the groups which are involved in their classification there are groups which possess nontrivial partitions and these were also studied by Otto Kegel (and by Petr Kontorovich and by Guido Zappa and by many othe group theorists) long time ago with the methods of the partition theory.
More precisely, if \(G\) is a finite group and the set of all prime divisors of the order of \(G\) is \(\pi(G)\), the “Gruenberg-Kegel graph” (also known in the literature with the name of “prime graph”) is the graph with vertices given by the elements of \(\pi(G)\) and two vertices \(p\) and \(q\) in \(\pi(G)\) are adjacent if and only if \(G\) contains an element of order \(pq\).
Some connected components of this graph were described very well by Maria Silvia Lucido in [M. S. Lucido, Rend. Semin. Mat. Univ. Padova 102, 1–22 (1999; Zbl 0941.20008)] and in some of her successive contributions. Essentially the role of the connected component of this graph containing the prime 2 is peculiar, but about the other components there isn’t a clear and uniform description of what happens in general (this is one of the “Problems” which are mentioned often in recent years and also in the paper under review by the authors).
It should be noted that the prime graph has been the subject of several generalizations, as illustrated by [F. G. Russo, “Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group”, Adv. Pure Math. 2, 391–396 (2012)], or by [A. F. Vasil’ev et al., Sib. Math. J. 63, No. 5, 849–861 (2022; Zbl 1516.20055); translation from Sib. Mat. Zh. 63, No. 5, 1010–1026 (2022)] or even in the context of infinite profinite groups by [W. Herfort et al., Topology Appl. 263, 26–43 (2019; Zbl 1442.22006)] and in many other contributions in group theory.
In the paper under review, the authors offer a description of the component of the prime graph focusing on the role of the prime 3 and considering the behaviour of the nonsolvable finite groups. It is interesting to note that among all the groups which are involved in their classification there are groups which possess nontrivial partitions and these were also studied by Otto Kegel (and by Petr Kontorovich and by Guido Zappa and by many othe group theorists) long time ago with the methods of the partition theory.
Reviewer: Francesco G. Russo (Rondebosch)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
References:
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| [11] | A.S. Kondrat’ev, N.A. Minigulov, “Finite groups without elements of order six”, Math. Notes, 104:5 (2018), 696-701 · Zbl 1425.20014 · doi:10.1134/S000143461811010X |
| [12] | M.S. Lucido, “Prime graph components of finite almost simple groups”, Rend. Sem. Mat. Univ. Padova, 102 (1999), 1-22 · Zbl 0941.20008 |
| [13] | W.B. Stewart, “Groups having strongly self-centralizing \(3\)-centralizers”, Proc. Lond. Math. Soc., III. Ser., 26:4 (1973), 653-680 · Zbl 0259.20014 · doi:10.1112/plms/s3-26.4.653 |
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