# Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$.

Problem:

Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$.

For $n=5,11,71$, $G$ exists.

Note: Note, that if Brocard's problem has infinite solution, it does not imply, that there are infinite $S_n$ with symmetric subgroup of order of $n^2-1$.

• It seems like this fails for $n \leq 4$ Commented Jul 11, 2016 at 13:06

It can't exist in general, in fact if $n=p-1$ where p is prime, then there does not even exist a sub-group of order $n^2-1$ because the order of a subgroup must divide that of the group and $\frac{(p-1)!}{(p-1)^2-1} = \frac{(p-1)!}{p(p-2)}$ is not an integer.
There are infinitely many positive integers $n$ such that $S_n$ has an abelian subgroup of order $n^2-1$.
Suppose $n$ is odd. Let $u\in\{-1,+1\}$ be such that $n\equiv u\pmod{4}$. Then, $$n^2-1=2^{k+1}\,\left(\frac{n-u}{2^k}\right)\,\left(\frac{n+u}{2}\right)\,,$$ where $2^k$ is the largest power of $2$ that divides $n-u$ (whence $k\geq 2$). If $$2^{k+1}+\left(\frac{n-u}{2^k}\right)+\left(\frac{n+u}{2}\right)\leq n\,,$$ then there exists an abelian subgroup of $S_n$ of order $n^2-1$, which is the product of three cyclic groups generated by cycles of length $2^{k+1}$, $\frac{n-u}{2^k}$, and $\frac{n+u}{2}$. Note that the inequality above is equivalent to $$n-u\geq\frac{2^{k+2}}{1-\frac{1}{2^{k-1}}}\,.$$ Hence, if $n-u=2^kq$, where $q\geq 9$ is an odd integer, then $S_n$ has a subgroup of order $n^2-1$. In other words, if $$n=2^k(2r+1)\pm1$$ where $k\geq 2$ and $r\geq 4$ are integers, then $S_n$ has a subgroup of order $n^2-1$.