Subgroup permutability degree of \(\mathrm{PSL}(2,p^n)\). (Q2845574)

Let \(G\) be a group, and write \(L(G)\) for its set of subgroups. The `subgroup permutability degree' of \(G\), \(\mathrm{spd}(G)\), was introduced by Tărnăuceanu and is defined as follows: NEWLINE\[NEWLINE\mathrm{spd}(G)=\frac{|\{(H,K)\in L(G)\times L(G)\mid HK=KH\}|}{|L(G)|^2}.NEWLINE\]NEWLINE In this paper the authors calculate \(\mathrm{spd}(G)\) for \(G=\mathrm{PSL}(2,p^n)\). To perform the calculation two principal facts are used. The first is the formula NEWLINE\[NEWLINE\mathrm{spd}(G)=\frac{1}{|L(G)|^2}\sum_{H\in L(G)} F_2(H),\tag{1}NEWLINE\]NEWLINE where we write \(F_2(H)\) to mean the number of factorizations of the group \(H\). (A `factorization' of \(H\) is a pair of subgroups \(A,B\leq H\) for which \(H=AB\).)NEWLINENEWLINE The second principal fact is that all subgroups of \(\mathrm{PSL}(2,p^n)\) are known (thanks to a theorem of Dickson) and, moreover, that there is a certain family of subgroups \(\{H_i,\;i\in I\}\), which `partition' \(\mathrm{PSL}(2,p^n)\), i.e. which satisfy NEWLINE\[NEWLINE\bigcup_{i\in I}H_i=\mathrm{PSL}(2,p^n)\quad\text{ and }\quad H_i\cap H_j=\{1\}\text{ for all }i\neq j.NEWLINE\]NEWLINE Now \(\mathrm{spd}(\mathrm{PSL}(2,p^n))\) is calculated by working through all subgroups \(H\) of \(\mathrm{PSL}(2,p^n)\), calculating \(F_2(H)\), and applying the formula (1). The resulting expression is not in closed form.NEWLINENEWLINE One final remark: the authors mention a number of results concerning factorizations of finite groups and state that ``the factorizations of a large variety of finite simple groups are known''. In fact the maximal factorizations of finite simple groups are completely understood, thanks to celebrated work of \textit{M. W. Liebeck, C. E. Praeger} and \textit{J. Saxl} [Mem. Am. Math. Soc. 432 (1990; Zbl 0703.20021)].