Products of conjugacy classes of two by two matrices (Q1904052)

The covering number \(\text{cn} (G)\) and the extended covering number \(\text{ecn} (G)\) of a simple noncommutative group is the least \(k\) such that \(\text{cn} (G)\leq k\) if \(C^k= G\) for every nontrivial conjugacy class \(C\) in \(G\), and \(\text{ecn} (G)\leq k\) if \(C_1\cdots C_k= G\) for any nontrivial conjugacy classes \(C_j\) in \(G\), respectively. The authors extend these notions to arbitrary groups: Let \(C_1, \dots, C_k\) be any conjugacy classes in \(G\) such that the normal subgroup generated by each of them contains \([G,G ]\). Then the extended covering number \(\text{ecn} (G)\) is the least \(k\) such that the product \(C_1\cdots C_k\) contains every similarity class \(C_0\) with \(C_0= C_1 \cdots C_k \bmod [G,G ]\), and the covering number \(\text{cn} (G)\) is the least \(k\) satisfying this condition for equal classes \(C_1= \dots= C_k\). Clearly, \(\text{cn} (G)\leq \text{ecn} (G)\). The authors compute the covering number and the extended covering number for the groups \(G= \text{PSL}_2 (F)\), \(\text{GL}_2 (F)\), \(\text{PGL}_2 (F)\), and \(\text{SL}_2 (F)\) over an arbitrary field \(F\). It turns out, that \(\text{cn} (G)\) and \(\text{ecn} (G)\) depend on certain properties of the field. So \(\text{cn} (G)\) and \(\text{ecn} (G)\) may be 2, 3, 4 or 5 depending on whether the field is quadratically closed, a \(C_1 (2)\)-field, or not a \(C_1 (2)\)-field.