Strong involutions in finite special linear groups of odd characteristic
In [J. Algebra 322, No. 3, 833--881 (2009; Zbl 1181.20044)], \textit{C. R. Leedham-Green} and \textit{E. A. O'Brien}, described an algorithm that constructs a generating set for classical groups over finite fields. The paper under review looks at this algorithm for \(\mathrm{SL}(n,q)\), and it is shown that the number of random elements required in this case is \(O(\log~n)\), which betters all known results. In the above-mentioned algorithm, a key step is to construct a strong involution \(t\) with \(\pm{1}\)-eigenspaces \(E_{\pm}\) whose dimension is between \(n/3\) and \(2n/3\). The main theorem of the paper asserts: There exist positive constants \(\kappa\) and \(n_0\) (which can be taken to be \(0.0002\) and \(n_0 = 700\) respectively) such that the following holds: if \(n \geq n_0\), \(t \in \mathrm{GL}(n,q)\) is a strong involution and \(g\) is a uniformly distributed random element of \(\mathrm{GL}(n,q)\), and \(z(g)\) is defined to be \(\mathrm{inv}(tt^g)\) (respectively \(I\)) according as to whether \(tt^g\) has even (respectively odd) order, then the restrictions of \(z(g)\) to the eigenspaces \(E_{\pm}(t)\) are strong involutions with probability at least \(k/\log(n)\). The proof uses properties of certain linear transformations over finite fields called ppd-elements (short for primitive prime divisors). The authors raise the question of proving an analogous theorem for other classical groups. Those interested in the subject matter of this paper may also be interested in looking at the paper [\textit{S. Bhunia} et al., Adv. Appl. Clifford Algebr. 30, No. 3, Paper No. 31, 23 p. (2020; Zbl 1458.20038)] which has some related but different type of results.
scientific article; zbMATH DE number 6834839