Groups generated by two \(p\)-cycles
The main result of this paper is a characterization of all permutation groups which can be generated by two \(p\)-cycles, \(p\) a prime. Let \(x\) and \(y\) be two \(p\)-cycles which together move \(n\) distinct points. Then either \(<x,y>=A_n\) or one of the following must occur: (i) \(n=p - <x,y>\cong Z_ p\), \(p=(r^ m-1)/(r-1)\) for some prime \(r\) and \(<x,y>\cong L_m(r)\), \(p=11\) and \(<x,y>\cong L_2(11)\) or \(M_{11}\), or \(p=23\) and \(<x,y>\cong M_{23}\), (ii) \(n=p+1 - <x,y>\cong L_2(p)\), \(p=11\) and \(<x,y>\cong M_{11}\), \(M_{12}\), or \(L_2(11)\), \(p=23\) and \(<x,y>\cong M_{24}\), or \(p\) is a Mersenne prime and \(<x,y>\) is a certain Frobenius group or a certain split extension of an elementary abelian \(2\)-group by \(L_m(2)\) where \(2^m=n\), (iii) \(n=3 - <x,y>\cong S_3\). The proof uses the characterization of simple 2-transitive groups due to \textit{P. J. Cameron} [Bull. Lond. Math. Soc. 13, 1--22 (1981; Zbl 0463.20003)].
scientific article; zbMATH DE number 4004444