## Cardinals of transitive permutation groups acting on $\{1,\dots,n\}$

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Is there a nice description of all divisors of $n!$ which can be realized as cardinals of permutation groups acting transitively on ${1,\dots,n}$?

A necessary condition is of course that such a divisor is a multiple of $n$.

$n$ (cyclic) and $2n$ (dihedral) are always possible but, without mistake on my behalf, $3n$ is not feasible unless $n\equiv 1\pmod 3$ (it can then be realized as a semi-direct product analogous to the dihedral group). can only be realized if $3$ divides the number of invertible integers modulo $n$.

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This is almost certainly too difficult to answer in general. What you write about $3n$ is not true. There are transitive groups of order $3n$ for $n=9$ and $n=14$ for example. On the other hand, there are none for $n=10$. The transitive groups have been enumerated for all $n \le 32$ by the way. Could you try asking a more specific question? – Derek Holt Apr 5 2012 at 12:23
Thank you for the correction: A semi-direct product needs of course divisibility by 3 of the number of invertible elements modulo $n$. I think you suggest that the answer is messy! – Roland Bacher Apr 5 2012 at 13:25

## 1 Answer

As Derek suggests in his comment, this question is too difficult to answer in general. However one could limit the question as follows: clearly if $K$ is a transitive permutation group then $|K|$ divides $|M|$ where $M$ is a maximal transitive subgroup of ${\mathrm Sym}(n)$; thus we can ask about the cardinality of a maximal transitive subgroup $M$ of ${\mathrm Sym}(n)$.

The O'Nan-Scott theorem is the main tool here. Roughly speaking it asserts that such a subgroup $M$ is either imprimitive (and hence a wreath product, with order formula easy), or else it is in a bunch of primitive families. Most of these families have a geometric description and, as such, it is easy to calculate their order.

The `difficult' family in this regard is the family of primitive almost simple groups. In this case one basically needs an enumeration of the maximal subgroups of all almost simple groups, which is a difficult problem but one which has received a great deal of attention.

Depending on what level of information you need, there are complete enumerations of maximal subgroups for many of the almost simples (although not all). However there are also some very nice general statements about the possible sizes of maximal subgroups. One example is this paper by Martin Liebeck; there are many others like it (many by Liebeck and his collaborators).

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