%\magnification=\magstep1
\input amstex
\documentstyle{amsppt}
\topmatter
\NoBlackBoxes
\title The Largest Prime Dividing the Maximal Order \\
of an Element of $S_n$ \endtitle
\affil Department of Mathematics \\
       University of Georgia \\
       Athens, GA  30602 
\endaffil
\thanks
A portion of this research, including the computations, was done at
the Supercomputing Research Center.  The author would also like to
thank the referee for helpful suggestions.
\endthanks
\abstract   We define $g(n)$ to be the maximal order of an element of
the symmetric group on $n$ elements.  Results about the prime factorization of
$g(n)$ allow a reduction of the upper bound on the largest prime
divisor of $g(n)$ to $1.328\sqrt{n\log n}$.
\endabstract
\email grantham\@joe.math.uga.edu \endemail
\author Jon Grantham \endauthor
\subjclass 20B40 \endsubjclass
\define\landau{1}
\define\mass{2}
\define\mnr{3}
\define\nic{4}
\define\nica{5}
\define\nicb{6}
\define\rands{7}
\define\schoen{8}
\endtopmatter
\document

Let $S_n$ be the symmetric group on $n$ letters.  
\proclaim{Definition}
$g(n)=\operatorname{max}\,\{\operatorname{ord}(\sigma)\,|\,\sigma\in S_n\}.$
\endproclaim

The first work on $g(n)$ was done by Landau \cite{\landau}
in 1903.  He showed that $\log{g(n)} \sim \sqrt{n\log n}$ as
$n\to\infty$.
In 1984, 
Massias \cite{\mass} showed an upper bound for $\frac{\log g(n)}%
{\sqrt{n\log n}}$,
$$ \log g(n) \le a\sqrt{n\log n}\qquad a = 1.05313\dots\qquad n\ge 1,$$
with $a$ attained for $n = 1,319,166$.

Let $P(g(n))$ be the largest prime divisor of $g(n)$. In 1969,
Nicolas \cite{\nic}
proved that $P(g(n)) \sim \sqrt{n\log n}$ as $n\to\infty$.
In 1989, Massias, Nicolas, and Robin \cite{\mnr} showed that $P(g(n))\le
2.86\sqrt{n\log n}$, $n\ge 2$.
They conjectured that $\frac{P(g(n))}{\sqrt{n\log n}}$
achieves a maximum ($1.265\dots$) for $n\ge 5$ at $n=215$, with $P(g(215))=43$.
They note that improving this bound using the 
techniques of their proof would require ``very extensive computation,''
and even then would not be able to reduce the constant in the bound
below $2$.

Using different techniques, however, we can improve this result to the
following

\proclaim{Theorem}
For each integer $n\ge 5$, we have
$$P(g(n))\le 1.328 \sqrt{n\log n}.$$
\endproclaim

Our proof begins with the simple observation that
$g(n)=\operatorname{max}\,\{\operatorname{ord}(\sigma)\,|\,\sigma\in S_n'\},$
where $S_n'$ is the subset of $S_n$ consisting 
of elements that are the product of disjoint cycles of prime power length.

To see this, recall the fact that we can write any $\sigma\in S_n$ as the
product of disjoint cycles.  Then $\operatorname{ord}(\sigma)$ is the least
common multiple of the cycle lengths. Consider a cycle of 
length $ab$ with $(a,b)=1$, $a,b>1$. The product of a cycle of length
$a$ with one of length $b$ also has order $ab$ and is a permutation on
fewer elements.  Thus, given any element of $S_n$, we may find
another that has the same order and is a product of disjoint cycles of prime
power length.

\proclaim{Definition}
For each natural number $M$, let $\ell(M)=\sum\limits_{p^\alpha\|M}{p^\alpha}.$
\endproclaim
We observe that $\ell(M)$ is the shortest length of a permutation of
order $M$.
Thus, we can characterize $g(n)$ in terms of $\ell$ as follows:
$$g(n)=\text{max}\,\{M\,|\,\ell(M)\le n\}.$$
In particular, $\ell(g(n))\le n$.

Nicolas \cite{\nicb} describes an algorithm for computing $g(n)$.
Employing a variation of this algorithm, I computed exact values of $g(n)$ 
for $n\le 500,000$ on a Sun 4/390.  The accuracy of
the computation was checked by calculating values of $g(n)$ using the set
$G$ described in \cite{\mnr} and verifying that they matched
those in the computations.  Analysis of the computations confirmed that
for $5\le n\le 500,000$, $\frac{P(g(n))}{\sqrt{n\log{n}}}$ attains a
maximum at $n=215$.

\proclaim{Lemma 1 (Nicolas \cite{\nica})} 
Let $p$, $p'$, and $q$ be distinct primes, with $q \ge p+p'$.  If $q$
divides $g(n)$, then at least one of $p$ and $p'$ divides $g(n)$.
\endproclaim
\demo{Proof}
Suppose $p$ and $p'$ are primes not dividing $g(n)$.
Assume there is a prime $q\ge{p+p'}$ with $q\mid g(n)$.
Without loss of generality, $p<p'$.  Choose $k$ such that
$$p^k+p'\leq q \le p^{k+1}+p'-1.$$
Let $M=\frac{p^kp'g(n)}q$.  Since $q\mid g(n)$, $M$ is an integer.  Then
$$\ell(M)\le\ell(g(n))+(p^k+p'-q)\leq\ell(g(n))\leq n.$$
Thus, an element of order $M$ can be written as a permutation on $n$
letters.
Also,
$$ \multline
p^kp'-q\ge p^kp'-p^{k+1}-p'+1=p^k(p'-p)-p'+1 \\
\ge p(p'-p)-p'+1=(p-1)(p'-p-1)\ge0.
\endmultline$$
Therefore, $p^kp'>q$, so $M>g(n)$.  But $g(n)$ is the maximal order of a
permutation on $n$ letters.  Thus, we have a contradiction, and the lemma is
proven. %
\enddemo

Write $q=P(g(n))$.  We immediately get the following
\proclaim{Corollary}
At most one prime less than $\frac q2$ fails to divide $g(n)$.
\endproclaim

\proclaim{Lemma 2}
Suppose $0<\alpha<\beta<1$.  
If at least one prime in the interval $(\alpha q,\beta q)$
divides $g(n)$, then at most one prime in the interval
$(\sqrt\beta q, \frac{(1+\alpha)q}2)$ fails to divide $g(n)$.
\endproclaim

\demo{Proof}
If two primes in the interval $(\sqrt\beta q, \frac{(1+\alpha)q}2)$ fail
to divide $g(n)$, call them $p$ and $p'$.  Let $q'$ be a prime in the
interval $(\alpha q,\beta q)$ dividing $g(n)$.  Let
$M=\frac{pp'}{qq'}g(n)$.  Then
$$\ell(M)\le p+p'-q-q'+\ell(g(n))\le (1+\alpha)q-q-\alpha q+\ell(g(n))%
=\ell(g(n)).$$
But $pp'-qq'>(\sqrt\beta q)^2-q(\beta q)=0,$ so $M>g(n)$, giving a
contradiction.
\enddemo

\demo{Proof of Theorem}
By the computations, we may take $n>500\text{,}000$.  We may also
assume $q\ge 1.3\sqrt{500000\log 500000}>3329$.   Using the results of
Schoenfeld \cite{\schoen} for large $q$, and computations for small $q>3329$, we
see that there are always at least two primes in the
intervals  $(\alpha_iq,\beta_iq)$, with $\alpha_1=.2426$,
$\beta_1=.25$, $\alpha_2=.3746$, $\beta_2=.386$, 
$\alpha_3=.4632$, $\beta_3=.4723$,
$\alpha_4=.5248$, $\beta_4=.5352$,
$\alpha_5=.57$, $\beta_5=.5812$,
$\alpha_6=.6044$, $\beta_6=.6162$,
$\alpha_7=.6312$, $\beta_7=.6435$,
$\alpha_8=.6534$, $\beta_8=.6652$, and
$\alpha_9=.6714$, $\beta_9=.6834$.
By Lemma 1, at most one of the two or more primes in any of the first three intervals
fails to divide $g(n)$.
Applying Lemma 2, we get that at most one prime in each interval 
$(\sqrt{\beta_i}q,\frac{(1+\alpha_i)q}2)$ fails to divide $g(n)$, for
$i\le 3$.
This fact in turn implies that at most one prime in each interval $(\alpha_i
q, \beta_iq)$ fails to divide $g(n)$ for $4\le i\le 9$.  Applying Lemma 2
again, we see that at most one prime in each interval
$(\sqrt{\beta_i}q,\frac{(1+\alpha_i)q}2)$ fails to divide $g(n)$ for
$1\le i\le 9$.

We note that these intervals cover $(.5q,.8357q)$.  So at most ten
primes less than $.8357q$ fail to divide $g(n)$, and they can be at most
$\frac q2$, $\frac{(1+\alpha_1)q}2$,\dots, and $\frac{(1+\alpha_9)q}2$.

Therefore,
$$g(n)\ge \frac{q\prod\limits_{p\le .8357q}p}%
{\frac q2\prod {\frac{1+\alpha_i}2q}}.$$
Taking logarithms, we get
$$\log{g(n)}\ge \theta(.8357q)-\log{\dfrac 12}-\sum{\log\left(%
\frac{1+\alpha_i}2q\right)},$$
where $\theta$ is the Chebyshev function,
$\theta(x)=\sum\limits_{p\le x}{\log p}$.

For $q>3329$ the sum of the terms on the right is less than $.01338q$, so
$$\log{g(n)}\ge \theta(.8357q)-.01338q.$$
Using the estimates for $\theta(x)$ in \cite{\rands}, we get
$$\log{g(n)}\ge .79307q.$$
From \cite{\mnr}, $1.05314\sqrt{n\log n}\ge \log{g(n)}$, so
$$1.328\sqrt{n\log n} \ge \frac{1.05314}{.79307}\sqrt{n\log n} \ge q.$$

It is likely that further computation would be able to show that 
$\frac{P(g(n))}{\sqrt{n\log{n}}}$ attains a
maximum at $n=215$ for \bf{all} $n\ge 5$.

\enddemo

\Refs

\ref\no1
\by E. Landau
\paper \"Uber die Maximalordnung der Permutationen gegebenen Grades
\jour Archiv der Math. und Phys.
\yr 1903
\pages 92--103
\endref

\ref\no2
\by J. P. Massias
\paper Majoration explicite de l'ordre maximum d'un \'el\'ement du
groupe sym\'etrique
\jour Ann. Fac. Sci. Toulouse Math.
\yr 1984
\vol 6
\pages 269--280
\endref

\ref\no3
\by J. P. Massias, J. L. Nicolas and G. Robin
\paper Effective Bounds for the Maximal Order of an Element in the Symmetric
Group
\jour Math. Comp.
\yr 1989
\vol 53
\pages 665--678
\endref

\ref\no4
\by J. L. Nicolas
\paper Sur l'ordre maximum d'un \'el\'ement dans le groupe $S_n$ des permutations
\yr1968
\jour Acta Arith.
\vol 14
\pages 315--332
\endref

\ref\no5
\by J. L. Nicolas
\paper Ordre maximal d'un \'el\'ement du groupe des permutations et
``highly composite numbers''
\jour Bull. Soc. Math. France
\vol 97
\yr 1969
\pages 129--191
\endref

\ref\no6
\by J. L. Nicolas
\paper Calcul de l'Ordre Maximum d'un \'El\'ement du Groupe Sym\'etrique $S_n$
\jour R.A.I.R.O.
\vol 3
\yr 1969
\pages 43--50
\endref

\ref\no7
\by J. B. Rosser and L. Schoenfeld
\paper Approximate Formulas for Some Functions of Prime Numbers
\jour Illinois J. Math.
\vol 6
\yr 1962
\pages 64--94
\endref

\ref\no8
\by L. Schoenfeld
\paper Sharper Bounds for the Chebyshev Functions $\theta(x)$ and 
$\psi(x)$.  II
\jour Math. Comp.
\vol 30
\yr 1976
\pages 337--360
\endref
\endRefs
\enddocument
