On the generation of finite simple groups by pairs of subgroups
Let A and B be finite non-trivial groups. The authors prove that for `almost all' pairs A, B there is a finite simple group which is generated by A and B. More precisely: Theorem 1. If \(| A| | B| \geq 12\) and q is any prime power, then for all \(n\geq 4| A| | B| +12\), the group \(G=PSL_ n(q)\) has subgroups \(\bar A\cong A\) and \(\bar B\cong B\) such that \(G=<\bar A,\bar B>\). - Theorem 2. If \(| A| | B| \geq 12\), then for all \(n\geq 4| A| | B| +12\) the alternating group \(A_ n\) has subgroups \(\bar A\cong A\) and \(\bar B\cong B\) such that \(A_ n=<\bar A,\bar B>.\) These two theorems have natural corollaries for generation of \(PSL_ n(q)\) and \(A_ n\) by pairs of elements. As an illustration of their general method the authors give a short proof that if \(n\geq 7\) then every group \(PSL_ n(q)\) can be generated by an involution and an element of order n-1.
generation of \(PSL_ n(q)\) and \(A_ n\) by pairs of elements
scientific article; zbMATH DE number 4057777