Characterizability of the group \(^2D_p(3)\) by its order components, where \(p\geq 5\) is a prime number not of the form \(2^m+1\).
Let \(G\) be a finite group and \(\pi_i\) (\(1\leq i\leq s\)), (\(s=s(G)\)), be all prime graph components of \(G\). Then \(|G|=m_1\cdots m_s\) for some coprime positive integers \(m_1,\dots,m_s\) such that \(\pi(m_i)=\pi _i\) (\(1\leq i\leq s\)). The integers \(m_1,\dots,m_s\) are called the order components of \(G\). The set \(\{m_1,\dots,m_s\}\) is denoted by \(\text{OC}(G)\). Many finite simple groups \(H\) with \(s(H)\geq 2\) are determined by \(\text{OC}(H)\). In this paper, the author proves that if \(G\) is a finite group and \(\text{OC}(G)=\text{OC}(^2D_p(3))\), where \(p\) is a prime with \(5\leq p\neq 2^m+1\), then \(G\) is isomorphic to \(^2D_p(3)\). A main consequence of this result is the validity of Thompson's conjecture for the groups under consideration.
scientific article; zbMATH DE number 5323451
scientific article