On Plotkin's Question on Two-Variable Identities Characterizing Finite Soluble Groups: Two Infinite Families of PSL(2,p) and Finite Certificates

B. I. Plotkin asked in the Kourovka Notebook (Problem 15.75 b), 2002) whether a finite group is soluble if and only if it satisfies one of the identities \(u_n=1\), where \(u_1=[x,y]\) and \(u_{n+1}=[[u_n,x],[u_n,y]]\). We give a partial answer: the question reduces to the minimal simple groups; \(\mathrm{PSL}(2,p)\) is settled for all primes \(p\equiv\pm1\pmod{16}\) and, in the Notebook's commutator convention, for all primes \(p\equiv0,\pm1\pmod7\), and every minimal simple group of order less than \(|\mathrm{PSL}(2,10007)|\) has a pair whose sequence never reaches \(1\); the question stays open. The reduction to Thompson's list of minimal simple groups is standard: it was outlined for this sequence by Nikolova and Plotkin (2001) and carried out by Bandman, Greuel, Grunewald, Kunyavskii, Pfister and Plotkin for another sequence. The families are explicit: if \(x,y\in\mathrm{SL}(2,p)\), \(\operatorname{tr} x=0\) and \(\operatorname{tr} y=\operatorname{tr} xy\) is a root of \(2t^4-4t^2+1\), then \(u_3=-u_2\ne\pm1\) (in the other commutator convention \(u_4=u_3\ne\pm1\)); if \(\operatorname{tr} x=1\) and \(\operatorname{tr} y=\operatorname{tr} xy\) is a root of \(t^3\pm t^2-2t\mp1\), then \(u_4=u_2\ne\pm1\) (in the Notebook's convention). By Dirichlet's theorem they contain half of the primes \(p>3\) with \(p\equiv\pm2\pmod5\), for which \(\mathrm{PSL}(2,p)\) is minimal simple. Checked finite certificates, pairs with \(u_M=u_{M+L}\ne1\), settle \(\mathrm{PSL}(3,3)\), the \(626\) groups \(\mathrm{PSL}(2,q)\) of Thompson's list with \(q\le10^4\), and \(\mathrm{Sz}(8)\), \(\mathrm{Sz}(32)\), \(\mathrm{Sz}(128)\); for \(\mathrm{PSL}(3,3)\) and \(\mathrm{PSL}(2,p)\), \(5\le p<100\), Nikolova and Plotkin had reported such pairs, found by computer. Not decided are the other groups of Thompson's list: \(\mathrm{PSL}(2,p)\) for primes \(p>10^4\) outside the families, \(\mathrm{PSL}(2,2^r)\) for primes \(r\ge17\), and \(\mathrm{PSL}(2,3^r)\) and \(\mathrm{Sz}(2^r)\) for primes \(r\ge11\). This is computational progress and two infinite families, not a near-solution. No printed answer to the question was found in the sources searched; we claim no priority. This is an unrefereed note.