Involutions in finite simple groups as products of conjugates (Q6594874)

The authors prove the following result: \N\NLet \(G\) be one of the classical groups \(GL^{\pm}_n(q)\), excluding \(GL^{\pm}_2(2)\) and \(GL_3(2)\), or \(Sp_{2n}(q)\), \(n \geq 2\), or \(O^{\pm}_n(q)\), \(n \geq 7\) and \(S = SL_n^{\pm}(q)\), \(Sp_{2n}(q)\) or \(\Omega^{\pm}_n(q)\), respectively. Let \(g \in G \setminus Z(G)\) and \(C = g^G\) be the conjugacy class. Then in all cases there is an involution \(t \in S\), which is a product of six terms \(x_iy_i^{-1}\) with \(x_i,y_i \in C\). This means that there are \(12\) elements \(x \in C \cup C^{-1}\) such that there product is an involution in the simple group.\N\NThey also prove a corresponding result for alternating groups, where we get a product of two terms \(x_iy_i^{-1}\) giving an involution besides \(n = 5\) and \(g\) is a 5-cycle. This is due to the fact that in the generic case there are \(g,h \in C\) with \([g,h]\) an involution.\N\NThe corresponding result for the sporadic groups has been proved by \textit{I. Zisser} [Isr. J. Math. 67, No. 2, 217--224 (1989; Zbl 0693.20016)] which yields an upper bound of \(6\) such products, while for the exceptional groups of Lie type, \textit{R. Lawther} and \textit{M. W. Liebeck} [J. Comb. Theory, Ser. A 83, No. 1, 118--137 (1998; Zbl 0911.05035)] showed that there is an upper bound of \(376\) such pairs.