A recognition of simple groups \(\text{PSL}(3,q)\) by their element orders. (Q1431122)

For a finite group \(G\), let \(\pi_e(G)\) be the spectrum of \(G\), i.e. the subset of the set of natural numbers consisting of the orders of elements in \(G\). A group \(G\) is said to be recognizable (by the spectrum) if every finite group \(H\) with \(\pi_e(H)=\pi_e(G)\) is isomorphic to \(G\). The authors prove that the finite simple group \(L_3(q)\) is recognizable if \(3<q\equiv\pm 2\pmod 5\) and \((6,(q-1)/2)=1\). Reviewer's remark: The recognizability problem for all groups \(L_3(q)\) is currently completely solved by \textit{A. V. Zavarnitsine} [Sib. Mat. Zh. 45, No. 2, 319-328 (2004)].