Primitive permutation group

Question

Consider the following result (Lemma 4.4.4 from "Topics in Galois Theory" byJean-Pierre Serre, Course at Harvard University, Fall 1988).

Lemma: Let $G$ be a transitive subgroup of $S_n$ which is generated by cycles of prime orders. Then:

1. $G$ is primitive.
2. If $G$ contains a transposition, then $G = S_n.$
3. If $G$ contains a $3$-cycle, then $G = A_n$ or $S_n.$

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Does it mean

1. "generated by cycles of prime order"? Iimplies the inertia group at any ramified prime is of prime order? Kindly help.

Answer 1

I guess you are assuming $G$ is transitive. Otherwise there are obvious counterexamples, as indicated in the comments.

There are also transitive counterexamples. For example, the maximal imprimitive group $G = S_3 \wr S_3 = S_3^3 \rtimes S_3 \le S_9$ is generated by a transposition and a $9$-cycle.