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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ Mar 17, 2023 at 13:41
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    $\begingroup$ Yes, except that the group $p:(p-1)$ is $2$-transitive! $\endgroup$
    – Derek Holt
    Mar 17, 2023 at 14:05
  • $\begingroup$ You're right I should have said $ n $ properly dividing $ p-1 $, that would be right! $\endgroup$ Mar 17, 2023 at 14:53

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