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Several quantitative characterizations of some specific groups. (English) Zbl 1424.20028

Summary: Let \(G\) be a finite group and let \(\pi(G)=\{p_1,p_2,\dots,p_k\}\) be the set of prime divisors of \(|G|\) for which \(p_1<p_2<\cdots <p_k\). The Gruenberg-Kegel graph of \(G\), denoted \(\operatorname{GK}(G)\), is defined as follows: its vertex set is \(\pi(G)\) and two different vertices \(p_i\) and \(p_j\) are adjacent by an edge if and only if \(G\) contains an element of order \(p_i p_j\). The degree of a vertex \(p_i\) in \(\text{GK}(G)\) is denoted by \(d_G(p_i)\) and the \(k\)-tuple \(D(G)=(d_G(p_1),d_G(p_2),\dots,d_G(p_k))\) is said to be the degree pattern of \(G\). Moreover, if \(\omega\subseteq\pi(G)\) is the vertex set of a connected component of \(\text{GK}(G)\), then the largest \(\omega\)-number which divides \(|G|\), is said to be an order component of \(\text{GK}(G)\). We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as \(U_4(2)\). Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as \(U_5(2)\).

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D05 Finite simple groups and their classification
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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