## 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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