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Finite groups with a five-component prime graph. (English. Russian original) Zbl 1275.20009

Sib. Math. J. 54, No. 1, 40-46 (2013); translation from Sib. Mat. Zh. 54, No. 1, 57-64 (2013).
From the introduction: Some classification is obtained for the finite groups whose prime graph has five connected components. In particular, we show that such a group is simple. Some related results are stated about representations of simple groups.
The prime graph \(\Gamma(G)\) of a finite group \(G\), also known as the Gruenberg-Kegel graph, is the graph whose vertex set is the set \(\pi(G)\) of prime divisors of the order \(|G|\) in which two distinct vertices \(p,q\in\pi(G)\) are joined by an edge if and only if \(G\) contains an element of order \(pq\). Let \(s(G)\) denote the number of connected components of \(\Gamma(G)\). In this paper, we show that the groups with \(s(G)=5\) admit classification.
The principal result is Theorem 1. Let \(G\) be a finite group with \(s(G)=5\). Then \(G\) is a simple group isomorphic to \(E_8(q)\), where \(q\equiv 0,\pm1\pmod5\).
In order to prove Theorem 1, we had to obtain some results about representations of simple groups. One of these results, stated below, may be of interest in its own right.
Proposition 2. Let \(G={^3D_4(q)}\) act on a nonzero vector space \(V\) over a field of characteristic not dividing \(q\) (possibly, zero). Then each element of \(G\) of order \(q^4-q^2+1\) fixes on \(V\) a nonzero vector.

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20C33 Representations of finite groups of Lie type
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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