Possible orders of two generators of the alternating and of the symmetric group

Abstract

It is well known that every alternating and every symmetric group can be generated by two of its substitutions, and that two such generating substitutions can usually be selected in a large number of different ways. Since two operators of order 2 must always generate a dihedral group it is evident that no alternating group can be generated by two of its substitutions of order 2, and that the only symmetric group which can be thus generated is the symmetric group of order 6. On the other hand, it is known that with very few exceptions, relating to groups whose degrees do not exceed 8, every alternating group and every symmetric group can be generated by two of its substitutions of orders 2 and 3 respectively.t In the present article we shall prove that whenever an alternating group involves a substitution of order 1>3 then it contains two substitutions of orders 2 and I respectively which generate the entire group. We shall also determine the degrees of all the symmetric groups to which a similar theorem does not apply. Before proving this general theorem, it may be desirable to consider the more elementary question of generating an alternating or a symmetric group by two of its substitutions which are separately composed of a single cycle. When neither of the two numbers 11, 12 exceeds n but their sum exceeds n it is obvious that two substitutions sl, s2 which are separately composed of a single cycle, and whose orders are 11, 12 respectively, can be so selected that they generate a transitive group of degree n, and that half the substitutions of this transitive group are negative whenever at least one of the two numbers 11, 12 iS even. If s1 and s2 do not have all their letters in common we may suppose that the common letters are arranged in the same order in both of these substitutions and hence their commutator is either of the form abc or of the form ab cd. If both of them are of degree n we may suppose that all their letters are arranged in the same order with the exception that two adjacent letters are interchanged. Hence their commutator is of the form abc in this case. The group generated by si, S2 is obviously multiply transitive