In the process of solving some exercise, I became curious about the maximum height of a chain of subgroups in $S_n$.
More specifically - what is the maximum length k of a chain of subgroups $\{e\} \subset H_1 \subset \ldots \subset H_k = S_n$, where no additional conditions are imposed on the subgroups other than the inclusions being proper.
Does anyone know any results in this direction? It feels somewhat beyond my current abilities.
My google searches for maximal series connected me to results about composition series, chief series, which I am already familiar with. Is there some reason why a maximal series of subgroups (in a finite group) is uninteresting in general? (I guess they are certainly less interesting than their normal series cousins, since the difference between successive terms cannot be measured by a group...)
Thanks in advance.
This value is known. The length of the longest subgroup chain in $S_{n}$ is given by $$\left\lceil\frac{3n}{2}\right\rceil - b(n) - 1,$$ where $b(n)$ is the number of $1$s in the base $2$ representation of $n$.
Reference: P.J. Cameron, R. Solomon and A. Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989), 340-352.
There is also an earlier paper:
Reference: L. Babai, On the length of subgroup chains in the symmetric group, Comm. Algebra 14 (1986), 1729-1736,
in which the upper bound $2n-1$ is established.
One reason why such a longest chain is interesting is that its length provides a bound on the minimal number of generators for any subgroup. In particular, for the case of $S_{n}$, the bound above provides an upper bound for the minimal number of generators of any permutation group of degree $n$.