# Are transitive permutation groups of prime degree 2-generated?

According to

Primitive permutation groups that are 2-generated

all 2-transitive permutation groups are 2-generated. There are, however, primitive groups of non prime degree, for example $$PrimitiveGroup(25,11)$$ for degree $$n=25$$, which are not $$2$$-generated.

What about for prime degree? Is it true that every transitive permutation group of prime degree is $$2$$-generated?

Yes, there is a theorem of Burnside that a simply transitive (i.e. transitive but not 2-transitive) permutation group $$G$$ of prime degree $$p$$ has a normal subgroup $$P$$ of order $$p$$, so $$G \le P \rtimes H$$ with $$H \le {\rm Aut}(P)$$. But $${\rm Aut}(P)$$ is cyclic of order $$p-1$$ so $$G$$ is $$2$$-generated.

• ah I see so the simply transitive permutation groups of prime degree are exactly the semidirect products of the form $p:n$ for $n$ dividing $p-1$. And all the simply transitive groups of prime degree are primitive. Mar 17, 2023 at 13:41
• Yes, except that the group $p:(p-1)$ is $2$-transitive! Mar 17, 2023 at 14:05
• You're right I should have said $n$ properly dividing $p-1$, that would be right! Mar 17, 2023 at 14:53