how many abelian transitive subgroups of $S_{n}$

Question

It's well-known that any abelian transitive subgroup A of a symmetric group $S_{n}$ has order $n$. Moreover, does anyone know how many abelian transitive subgroups of $S_{n}$ and what do they look like?

Answer 1

Every abelian group $A$ of order $n$ acts transitively on itself by left translation (aka Cayley action). Therefore we can view any such $A$ as a transitive subgroup of $S_n$. Furthermore, given that this kind of an action necessarily has trivial stabilizers, such an action has to be isomorphic to the translation action. In other words, a given abelian group $A$ of order $n$ has an essentially unique (up to conjugation by an element of $S_n$) transitive action of this type.

The problem of listing abelian groups of a given order $n$ (up to isomorphism) is relatively easy, but does depend on having the full factorization $n=\prod_{i=1}^kp_i^{a_i}$ with $p_i$ ranging over the prime factors of $n$. You need to list all the possible $p$-parts for $p=p_i,i=1,2,\ldots,k$. This amounts to partitioning $a_i$ in all possible ways, and the group is then a direct sum of its Sylow $p_i$-subgroups. The number of different (up to isomorphism) abelian groups of order $n$ is thus $$ \prod_{i=1}^kp(a_i), $$ where $p$ is the partition function.