Result: The Number of Isomorphism Classes of Finite Groups with Given Element Orders: The number of isomorphism classes of finite groups with given element orders

Let \(G\) be a finite group, let \(\pi_e(G)\) denote the set of orders of elements in \(G\), and let \(h(\pi_e(G))\) denote the number of pairwise nonisomorphic finite groups \(H\) with equality of the orders \(\pi_e(H)=\pi_e(G)\). The article provides a partial answer to problem 13.63 in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory, Institut Matematiki SO RAN, Novosibirsk (2002; Zbl 0999.20002)] on the existence of a natural number \(k\) with the property that \(h(\pi_e(G))\in\{1,2,\dots,k,\infty\}\) for every finite group \(G\). The Gruenberg-Kegel graph of a group \(G\) is the graph whose set of vertices is the set \(\pi(G)\) of all prime divisors of order the of \(G\); vertices \(p\), \(q\) are connected by an edge if and only if \(G\) contains an element of order \(pq\). Let \(t(G)\) be the number of connected components of the graph. Theorem. Let \(G\) be a finite group and let \(t(G)\geq 3\). Then \(h(\pi_e(G))\in\{1,\infty\}\). Conjecture. For every group \(G\) the containment \(h(\pi_e(G))\in\{1,2,\infty\}\) holds.