On Minimal Generating Sets for Symmetric and Alternating Groups

Abstract

By a famous result, the subgroup generated by the n-cycle $$\sigma =(1,2,\ldots ,n)$$ and the transposition $$\tau =(a,b)$$ is the full symmetric group $$S_{n}$$ if and only if $$gcd(n,b-a)=1$$ . In this paper, we first generalize the above result for one n-cycle and k arbitrary transpositions, and then provide similar necessary and sufficient conditions for the subgroups of $$S_{n}$$ in the following three cases: first, the subgroup generated by the n-cycle $$\sigma $$ and a 3-cycle $$\delta =(a,b,c)$$ , second, the subgroup generated by the n-cycle $$\sigma $$ and a set of transpositions and 3-cycles, and third, by the n-cycle $$\sigma $$ and an involution (a, b)(c, d). In the first case, we also determine the structure of the subgroup generated by $$(1,2,\ldots ,n)$$ and a 3-cycle $$\delta =(a,b,c)$$ in general. Finally, an application to unsolvability of a certain infinite family of polynomials by radicals is given.