Abstract:Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2,\ldots, p_k\}$ 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 ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots, d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi(G)$ is the vertex set of a connected component of $\operatorname{GK} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname{GK} (G)$. 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~$U_5(2)$.
Keywords: OD-characterization of finite group; prime graph; degree pattern; simple group; $2$-Frobenius group