A primitive subgroup of the symmetric group 
is equal to either the alternating group 
or 
whenever it contains at least one permutation which
is a 
-cycle
for some prime 
.
Jordan's Symmetric Group Theorem
See also
Symmetric GroupExplore with Wolfram|Alpha
References
Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199-205, 1969.Wielandt, H. Finite Permutation Groups. New York: Academic Press, 1964.Referenced on Wolfram|Alpha
Jordan's Symmetric Group TheoremCite this as:
Weisstein, Eric W. "Jordan's Symmetric Group Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JordansSymmetricGroupTheorem.html