Quasirecognition of a class of finite simple groups by the set of element orders.

*(Russian, English)*Zbl 1035.20012
Sib. Mat. Zh. 44, No. 2, 241-255 (2003); translation in Sib. Math. J. 44, No. 2, 195-207 (2003).

Given a finite group \(G\), denote by \(\omega(G)\) the set of its element orders. A finite group \(G\) is called recognizable by \(\omega(G)\) if every finite group \(H\) with \(\omega(H)=\omega(G)\) is isomorphic to \(G\). The authors investigate a weaker property. A finite group \(G\) is called quasirecognizable by \(\omega(G)\) if every finite group \(H\) with \(\omega(H)=\omega(G)\) has a section isomorphic to \(G\).

The Gruenberg-Kegel graph \(GK(G)\) of \(G\) is the graph in which the vertices are the prime divisors of \(| G|\) and two distinct vertices \(p\) and \(q\) are joined if and only if \(pq\in\omega(G)\). The authors [Ukr. Mat. Zh. 54, No. 7, 998-1003 (2002; Zbl 1024.20014)] proved that if \(L\) is a simple finite group for which the Gruenberg-Kegel graph has at least 4 connected components then \(L\) is quasirecognizable by \(\omega(L)\). In the article under review, they extend their result as follows: If \(L\) is a simple finite group for which the Gruenberg-Kegel graph has at least 3 connected components then \(L\) is quasirecognizable by \(\omega(L)\) except for the alternating group \(A_6\).

The Gruenberg-Kegel graph \(GK(G)\) of \(G\) is the graph in which the vertices are the prime divisors of \(| G|\) and two distinct vertices \(p\) and \(q\) are joined if and only if \(pq\in\omega(G)\). The authors [Ukr. Mat. Zh. 54, No. 7, 998-1003 (2002; Zbl 1024.20014)] proved that if \(L\) is a simple finite group for which the Gruenberg-Kegel graph has at least 4 connected components then \(L\) is quasirecognizable by \(\omega(L)\). In the article under review, they extend their result as follows: If \(L\) is a simple finite group for which the Gruenberg-Kegel graph has at least 3 connected components then \(L\) is quasirecognizable by \(\omega(L)\) except for the alternating group \(A_6\).

Reviewer: M. A. Grechkoseeva (Novosibirsk)