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\begin{center}
\vskip 1cm{\LARGE\bf Converse Lagrange Theorem Orders \\
\vskip .1in
and Supersolvable Orders
}
\vskip 1cm
\large
Des MacHale\\
School of Mathematical Sciences\\
University College Cork\\
Cork\\
Ireland\\
\href{mailto:d.machale@ucc.ie}{\tt d.machale@ucc.ie}
\\ \ \\
Joseph Manning\\
Department of Computer Science\\
University College Cork\\
Cork\\
Ireland\\
\href{mailto:manning@cs.ucc.ie}{\tt manning@cs.ucc.ie}
\end{center}

\vskip .2 in

\begin{abstract}
For finite groups, we investigate both converse Lagrange theorem (CLT) orders
and supersolvable (SS) orders, and see that the latter form a proper subset of
the former.  We focus on the difference between these two sets of orders,
reformulate the work of earlier authors algorithmically, and construct a
computer program to enumerate such NSS-CLT orders.  We establish several results
relating to NSS and CLT orders and, working from our computer-generated data,
propose a pair of conjectures and obtain a complete characterization of the most
common form of NSS-CLT order.
\end{abstract}

\section{Introduction}

Throughout this paper, we consider only \emph{finite} groups, and we begin by
recalling one of the most fundamental results in the area, the famous theorem of
Lagrange:

\begin{theorem}\label{thm:Lagrange}
If $G$ is a group and $H$ is a subgroup of $G$, then $|H|$ is a divisor of
$|G|$.
\end{theorem}

However, the converse of this result is false; for example, $A_4$, the
alternating group on four symbols, which has order 12, has no subgroup of order
6~\cite{BM}.

If $G$ is a group which has a subgroup of order $d$, for \emph{every} divisor
$d$ of $|G|$, then $G$ is a \emph{converse Lagrange theorem} (CLT) group;
otherwise, $G$ is a \emph{non-converse Lagrange theorem} (NCLT) group.  For
example, $S_4$, the symmetric group on four symbols, is CLT, since it has order
24 and has subgroups of order 1, 2, 3, 4, 6, 8, 12, and 24, all the divisors of
24.  As~mentioned above, $A_4$ is an NCLT group.

If \emph{every} group of order $n$ is CLT, then $n$ is a \emph{CLT order};
otherwise, $n$ is an \emph{NCLT order}.  For example, 16 is a CLT order, since
every group of order 16 has subgroups of order 1, 2, 4, 8, and 16.  On the other
hand, 24 is an NCLT order, because, although $S_4$ is a CLT group, another group
$SL(2,3)$ of order 24 has no subgroup of order 12~\cite{BM}.

We also consider the following concept.  A group $G$ is \emph{supersolvable}, or
\emph{supersoluble}, (SS) if it has a series of subgroups, each normal in the
next:
\[
   \{1\} = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \,\cdots\,
           \triangleleft H_{k-1} \triangleleft H_k = G
\]
satisfying both of the following conditions:
\begin{itemize}
\item $H_i \triangleleft G$, for each $0 \le i \le k$
\item $H_i / H_{i-1}$ is cyclic, for each $1 \le i \le k$.
\end{itemize}
For example, $D_4$, the group of rigid motions of a square, is SS, whereas the
group $S_4$ is NSS (non-supersolvable).

If \emph{every} group of order $n$ is SS, then $n$ is an \emph{SS order};
otherwise, $n$ is an \emph{NSS order}.  For~example, 8 is an SS order, whereas
24 is an NSS order since $S_4$ is an NSS group.

The concepts of CLT groups and SS groups are linked by the following result,
which has been proved by several authors (see Bray~\cite{Br}, for example):

\begin{theorem}\label{thm:SSisCLT}
Every SS group is CLT.
\end{theorem}

It follows immediately that \emph{every SS order is a CLT order}; restating this
in an equivalent but more useful form for what follows, we have

\begin{theorem}\label{thm:NCLTisNSS}
Every NCLT order is an NSS order.
\end{theorem}

The converse of Theorem~\ref{thm:SSisCLT} is false.  For example, $S_4$ is CLT
yet also NSS\@.  Nevertheless, the possibility remains that the NCLT
\emph{orders} and the NSS \emph{orders} coincide.  As just seen, $S_4$, of order
24, is CLT but NSS; however, there exists another group of order 24, $SL(2,3)$,
which is NCLT and NSS, making the integer 24 both an NSS order and an NCLT
order.  However, we shall shortly see that the NCLT orders and the NSS orders do
\emph{not} coincide, and so the converse of Theorem~\ref{thm:NCLTisNSS} is also
false.

Curran~\cite{Cu} has found all the NCLT \emph{groups} with order less than 100,
but a full structure theorem for the rather complicated class of all NCLT
groups currently seems out of reach.  However, the NCLT \emph{orders} have been
completely determined by Berger~\cite{Be}, although this classification is quite
involved, consisting of five different sets of numerical conditions relating to
the divisors of the group order and the congruences which they must satisfy.
Likewise, the NSS \emph{orders} have been determined by Pazderski~\cite{Pa}, and
while their classification is somewhat less complicated that that of the NCLT
orders, it is still quite intricate (Hughes~\cite{Hu} presents a simpler
formulation, and in English rather than in the German of Pazderski's paper).

\section{NSS-CLT orders}

We have reformulated these classifications of Berger and Pazderski\,/\,Hughes
algorithmically and implemented them on a computer, so as to view and compare
NSS\footnote{NSS orders also appear as sequence \seqnum{A066085} in the
\emph{On-Line Encyclopedia of Integer Sequences}~\cite{Sl}.} and NCLT orders.

Inspection of Table~\ref{tab:NSS-NCLT} gives rise to several results and
conjectures, as presented below.

\begin{theorem}\label{thm:NCLTsubNSS}
The NCLT orders form a proper subset of the NSS orders.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:NCLTisNSS} establishes inclusion, and the number 224 from
Table~\ref{tab:NSS-NCLT} below, being an NSS order but not an NCLT order, shows
that this inclusion is proper.
\end{proof}

The NSS-CLT orders (numbers which are NSS orders but not NCLT orders) form the
main focus of this paper.  The number \,$224 = 2^5 \D 7$\, is in fact the
smallest NSS-CLT order; the next such order is \,$2464 = 2^5 \D 7 \D 11$, while
the first such odd order is \,$3159 = 3^5 \D 13$.

\begin{theorem}\label{thm:NSSmult}
Every multiple of an NSS order is itself an NSS order.
\end{theorem}

\begin{proof}
Let $k$ be a positive integer and $n$ an NSS order.  Choose any NSS group of
order $n$, and consider its direct product with the cyclic group $C_k$.  The
resulting group has order $kn$, and is clearly NSS, as it has an NSS subgroup;
this follows from the fact that every subgroup of an SS group is SS\@.  Thus
$kn$ is an NSS order, as claimed.
\end{proof}

However, the situation is different in the case of NCLT orders:

\begin{theorem}\label{thm:noNCLTmult}
A multiple of an NCLT order need not be an NCLT order.
\end{theorem}

\begin{proof}
 From Table~\ref{tab:NSS-NCLT} below, a minimal counterexample is given by the
CLT order \,$224 = 2^5 \D 7$, which is a multiple of the NCLT order
\,$56 = 2^3 \D 7$.
\end{proof}

The minimal such odd order is the CLT order \,$3159 = 3^5 \D 13$, which is a
multiple of the NCLT order \,$351 = 3^3 \D 13$.  For completeness, we also note

\begin{theorem}\label{thm:noNSSCLTmult}
A multiple of an NSS-CLT order need not be an NSS-CLT order.
\end{theorem}

\begin{proof}
We have $2 \D 224 = 448$\,; but 224 is an NSS-CLT order, while 448 is not.
\end{proof}

\begin{center}
{\footnotesize{%
  \begin{tabular}{rclcc}
  \multicolumn{3}{c}{NSS Order} & & NCLT? \\
  \cline{1-3} \cline{5-5} \\[-1.2ex]
    \LINE {  12} {2^2 \D 3       } {\YES}
    \LINE {  24} {2^3 \D 3       } {\YES}
    \LINE {  36} {2^2 \D 3^2     } {\YES}
    \LINE {  48} {2^4 \D 3       } {\YES}
    \LINE {  56} {2^3 \D 7       } {\YES}
    \LINE {  60} {2^2 \D 3 \D 5  } {\YES}
    \LINE {  72} {2^3 \D 3^2     } {\YES}
    \LINE {  75} {3 \D 5^2       } {\YES}
    \LINE {  80} {2^4 \D 5       } {\YES}
    \LINE {  84} {2^2 \D 3 \D 7  } {\YES}
    \LINE {  96} {2^5 \D 3       } {\YES}
    \LINE { 108} {2^2 \D 3^3     } {\YES}
    \LINE { 112} {2^4 \D 7       } {\YES}
    \LINE { 120} {2^3 \D 3 \D 5  } {\YES}
    \LINE { 132} {2^2 \D 3 \D 11 } {\YES}
    \LINE { 144} {2^4 \D 3^2     } {\YES}
    \LINE { 150} {2 \D 3 \D 5^2  } {\YES}
    \LINE { 156} {2^2 \D 3 \D 13 } {\YES}
    \LINE { 160} {2^5 \D 5       } {\YES}
    \LINE { 168} {2^3 \D 3 \D 7  } {\YES}
    \LINE { 180} {2^2 \D 3^2 \D 5} {\YES}
    \LINE { 192} {2^6 \D 3       } {\YES}
    \LINE { 196} {2^2 \D 7^2     } {\YES}
    \LINE { 200} {2^3 \D 5^2     } {\YES}
    \LINE { 204} {2^2 \D 3 \D 17 } {\YES}
    \LINE { 216} {2^3 \D 3^3     } {\YES}
    \LINE { 224} {2^5 \D 7       } {\NO }
    \LINE { 225} {3^2 \D 5^2     } {\YES}
    \LINE { 228} {2^2 \D 3 \D 19 } {\YES}
    \LINE { 240} {2^4 \D 3 \D 5  } {\YES}
    \LINE { 252} {2^2 \D 3^2 \D 7} {\YES}
    \LINE { 264} {2^3 \D 3 \D 11 } {\YES}
    \LINE { 276} {2^2 \D 3 \D 23 } {\YES}
    \LINE { 280} {2^3 \D 5 \D 7  } {\YES}
    \LINE { 288} {2^5 \D 3^2     } {\YES}
    \LINE { 294} {2 \D 3 \D 7^2  } {\YES}
    \LINE { 300} {2^2 \D 3 \D 5^2} {\YES}
    \LINE { 312} {2^3 \D 3 \D 13 } {\YES}
    \GAP
    \LINE { 351} {3^3 \D 13      } {\YES}
    \LINE { 360} {2^3 \D 3^2 \D 5} {\YES}
  \end{tabular}
  \hfill
  \begin{tabular}{rclcc}
  \multicolumn{3}{c}{NSS Order} & & NCLT? \\
  \cline{1-3} \cline{5-5} \\[-1.2ex]
    \LINE { 363} {3 \D 11^2      } {\YES}
    \GAP
    \LINE { 448} {2^6 \D 7       } {\YES}
    \GAP
    \LINE { 672} {2^5 \D 3 \D 7  } {\YES}
    \GAP
    \LINE { 896} {2^7 \D 7       } {\YES}
    \GAP
    \LINE {1120} {2^5 \D 5 \D 7  } {\YES}
    \GAP
    \LINE {1344} {2^6 \D 3 \D 7  } {\YES}
    \GAP
    \LINE {1568} {2^5 \D 7^2     } {\YES}
    \GAP
    \LINE {1792} {2^8 \D 7       } {\YES}
    \GAP
    \LINE {2016} {2^5 \D 3^2 \D 7} {\YES}
    \GAP
    \LINE {2240} {2^6 \D 5 \D 7  } {\YES}
    \GAP
    \LINE {2464} {2^5 \D 7 \D 11 } {\NO }
    \GAP
    \LINE {2688} {2^7 \D 3 \D 7  } {\YES}
    \GAP
    \LINE {2912} {2^5 \D 7 \D 13 } {\NO }
    \GAP
    \LINE {3136} {2^6 \D 7^2     } {\YES}
    \GAP
    \LINE {3159} {3^5 \D 13      } {\NO }
    \GAP
    \LINE {3808} {2^5 \D 7 \D 17 } {\NO }
    \GAP
    \LINE {4032} {2^6 \D 3^2 \D 7} {\YES}
    \GAP
    \LINE {4256} {2^5 \D 7 \D 19 } {\NO }
    \GAP
    \LINE {4480} {2^7 \D 5 \D 7  } {\YES}
    \GAP
    \LINE {4704} {2^5 \D 3 \D 7^2} {\YES}
    \GAP
    \LINE {4928} {2^6 \D 7 \D 11 } {\YES}
    \GAP
    \LINE {5152} {2^5 \D 7 \D 23 } {\NO }
    \GAP
    \vspace*{-2.37ex}
  \end{tabular}
  \hfill
  \begin{tabular}{rclcc}
  \multicolumn{3}{c}{NSS Order} & & NCLT? \\
  \cline{1-3} \cline{5-5} \\[-1.2ex]
    \LINE {5376} {2^8 \D 3 \D 7      } {\YES}
    \GAP
    \LINE {5600} {2^5 \D 5^2 \D 7    } {\YES}
    \GAP
    \LINE {5824} {2^6 \D 7 \D 13     } {\YES}
    \GAP
    \LINE {6048} {2^5 \D 3^3 \D 7    } {\YES}
    \GAP
    \LINE {6272} {2^7 \D 7^2         } {\YES}
    \GAP
    \LINE {6318} {2 \D 3^5 \D 13     } {\NO }
    \GAP
    \LINE {6496} {2^5 \D 7 \D 29     } {\NO }
    \GAP
    \LINE {6720} {2^6 \D 3 \D 5 \D 7 } {\YES}
    \GAP
    \LINE {6944} {2^5 \D 7 \D 31     } {\YES}
    \GAP
    \LINE {7168} {2^{10} \D 7        } {\YES}
    \GAP
    \LINE {7392} {2^5 \D 3 \D 7 \D 11} {\YES}
    \GAP
    \LINE {7616} {2^6 \D 7 \D 17     } {\YES}
    \GAP
    \LINE {7840} {2^5 \D 5 \D 7^2    } {\YES}
    \GAP
    \LINE {8064} {2^7 \D 3^2 \D 7    } {\YES}
    \GAP
    \LINE {8288} {2^5 \D 7 \D 37     } {\NO }
    \GAP
    \LINE {8512} {2^6 \D 7 \D 19     } {\YES}
    \GAP
    \LINE {8736} {2^5 \D 3 \D 7 \D 13} {\YES}
    \GAP
    \LINE {8960} {2^8 \D 5 \D 7      } {\YES}
    \GAP
    \LINE {9184} {2^5 \D 7 \D 41     } {\NO }
    \GAP
    \LINE {9408} {2^6 \D 3 \D 7^2    } {\YES}
    \GAP
    \LINE {9632} {2^5 \D 7 \D 43     } {\NO }
    \GAP
    \LINE {9856} {2^7 \D 7 \D 11     } {\YES}
    \GAP
    \vspace*{-2.64ex}
  \end{tabular}}}
\captionof{table}{Some NSS orders and their NCLT status}
\label{tab:NSS-NCLT}
\end{center}

We note that the number 224 occurring in Theorems~\ref{thm:NCLTsubNSS}
and~\ref{thm:noNCLTmult} can be derived from another source --- the following
theorem of Struik~\cite{St} --- which, however, makes no mention of minimality:

\begin{theorem}\label{thm:Struik}
Let $p$ and $q$ be primes, such that $q | (p-1)$ and such that $f$, the exponent
of $q\:(\mathrm{mod}\ p)$, is odd.  Then for each $1 \le m < q$, the number
$q^{2f-1}\,p^m$ is an NSS-CLT order.
\end{theorem}

Choosing $p = 7$ and $q = 2$ gives $q | (p-1)$ and $f = 3$, which is odd.  The
only valid choice for $m$ is 1.  Thus
$\,q^{2f-1}\,p^m \,=\, 2^5 \D 7^1 \,=\, 224\,$ is an NSS-CLT order, as already
seen.

Note that Theorem~\ref{thm:Struik} does not generate \emph{all} NSS-CLT orders;
for example, the second smallest NSS-CLT order is \,$2464 = 2^5 \D 7 \D 11$, but
this has three distinct prime factors.

We also recall the following result of Humphreys and Johnson~\cite{HJ}:

\begin{theorem}\label{thm:cubefree-groups}
Every CLT group of cubefree order is an SS group.
\end{theorem}

The cubefree condition is necessary, as the NSS-CLT group $S_4$, of order
$24 = 2^3 \D 3$, shows.  Moving now from groups to orders in
Theorem~\ref{thm:cubefree-groups}, and presenting the contrapositive to better
align with our general exposition, yields

\begin{theorem}\label{thm:cubefree-orders}
Every cubefree NSS order is an NCLT order.
\end{theorem}

Our computer-generated list of NSS-CLT orders, which extends
Table~\ref{tab:NSS-NCLT} above, contains the number \,$453789 = 3^3 \D 7^5$,
giving

\begin{theorem}\label{thm:cubefree-necessary}
The cubefree condition in Theorem~\ref{thm:cubefree-orders} is necessary.
\end{theorem}

The following well-known result of Deskins~\cite{De} will be useful in the
sequel:

\begin{theorem}\label{thm:deskins}
If $G$ is CLT and every subgroup of $G$ is CLT, then $G$ is SS.
\end{theorem}

\begin{corollary}\label{cor:deskins}
If $n$ is a CLT order and every proper divisor of $n$ is a CLT order, then $n$
is an SS order.
\end{corollary}

We now introduce the concepts of NCLT orders and NSS orders being
\emph{primitive}.

\begin{definition}
A number is a \emph{primitive NCLT order} if it is an NCLT order but none of its
proper divisors is an NCLT order.
\end{definition}

For example, Table~\ref{tab:NSS-NCLT} above shows that the NCLT orders 12, 56,
and 75 are primitive, whereas \,$24 = 2 \D 12$\, is not.

\begin{definition}
A number is a \emph{primitive NSS order} if it is an NSS order but none of its
proper divisors is an NSS order.
\end{definition}

For example, Table~\ref{tab:NSS-NCLT} above shows that the NSS orders 12, 56,
and 75 are primitive, whereas \,$224 = 4 \D 56$\, is not.

\begin{theorem}\label{thm:prim-equal}
The primitive NCLT orders and the primitive NSS orders coincide.
\end{theorem}

\begin{proof}
Let $n$ be a primitive NCLT order.  From Theorem~\ref{thm:NCLTisNSS} we know
that $n$ is an NSS order, so we need to show that if $d$ is a proper divisor of
$n$, then every group $G$ of order $d$ is SS\@.  By hypothesis, $G$ is CLT;
moreover, every subgroup of $G$, whose order divides $d$ and thus $n$, is also
CLT\@.  By Theorem~\ref{thm:deskins}, $G$ is SS, and so $n$ is a primitive NSS
order.

Conversely, let $n$ now be a primitive NSS order.  Every proper divisor of $n$
is an SS order, and thus, from Theorem~\ref{thm:SSisCLT}, a CLT order.  If $n$
itself were a CLT order, then by Corollary~\ref{cor:deskins}, $n$ would be an SS
order, a contradiction.  So $n$ must be an NCLT order.  In fact, $n$ must be a
\emph{primitive} NCLT order, since otherwise there would exist an NCLT group $H$
whose order is a~proper divisor of $n$; but from Theorem~\ref{thm:SSisCLT}, $H$
is an NSS group, contradicting the assumption that $n$ is a primitive NSS order.
\end{proof}

\begin{lemma}\label{lem:224p-NSS-CLT}
For each prime $p > 224$, the number $224p$ is an NSS-CLT order.
\end{lemma}

\begin{proof}
 From Table~\ref{tab:NSS-NCLT} we observe that 224 is an NSS order, so it then
follows from Theorem~\ref{thm:NSSmult} that $224p$ is an NSS order.

To show that $224p$ is a CLT order for each prime $p > 224$, let $G$ be an
arbitrary group of order $n = 224p$ and let $n_p$ be the number of Sylow
$p$-subgroups of $G$\@.  By Sylow's theorems, $n_p$ must be a divisor of $n$,
and $n_p \equiv 1\:(\mathrm{mod}\ p)$.  Now the divisors of $n$ are
\begin{center}
\begin{tabular}{llllllllllll}
1, & 2, & 4, & 8, & 16, & 32, & 7, & 14, & 28, & 56, & 112, & 224, \\
$p$, & $2p$, & $4p$, & $8p$, & $16p$, & $32p$, & $7p$, & $14p$, & $28p$,
     & $56p$, & $112p$, & $224p$. \\
\end{tabular}
\end{center}
The condition $n_p \equiv 1\:(\mathrm{mod}\ p)$ eliminates the second line, and
the fact that $p > 224$ leaves $n_p = 1$ as the only possibility.  Again, from
Sylow's theorems it follows that $G$ has a unique Sylow $p$-subgroup $G_p$, and
that $G_p \triangleleft\, G$.  Clearly, $| G_p | = p$.

Now $| G / G_p | = 2^5 \D 7$.  By Burnside's famous
``\emph{pq} theorem''~\cite[ch.\ XV]{Bu}, $G / G_p$ is solvable.  Since $G_p$,
being a $p$-group, is also solvable, it then follows that $G$ itself is
solvable.

By Hall's theorem~\cite{Ha}, $G$ has a (Hall) subgroup $H$ of order $2^5 \D 7$.
As $2^5 \D 7$ is a CLT order (see Table~\ref{tab:NSS-NCLT}), $H$ has subgroups
of every order dividing $2^5 \D 7$; thus also:
\begin{equation}
G \mbox{~has subgroups of orders~} 1, 2, 4, 8, 16, 32, 7, 14, 28, 56, 112, 224.
\label{eq:one}
\end{equation}

As noted above, $| G / G_p | = 2^5 \D 7$, which is a CLT order, so $G / G_p$ has
subgroups of orders 1, 2, 4, 8, 16, 32, 7, 14, 28, 56, 112, 224.  But each
subgroup of $G / G_p$ has the form $K / G_p$, for $K$ a subgroup of $G$, and
since $| K / G_p | = | K | / p$, it follows that:
\begin{equation}
G \mbox{~has subgroups of orders~} p, 2p, 4p, 8p, 16p, 32p, 7p, 14p, 28p, 56p,
112p, 224p.
\label{eq:two}
\end{equation}

Combining (\ref{eq:one}) and (\ref{eq:two}) shows that $G$ is a CLT group, and
so $224p$ is a CLT order.
\end{proof}

Note from Table~\ref{tab:NSS-NCLT} that this result does \emph{not} hold for any
of the primes $p = 2, 3, 5, 7, 31$.  But inspecting an extended version of this
table, as generated by our computer program, reveals that these are the
\emph{only} exceptions in the range $p \le 224$, giving

\begin{theorem}\label{thm:224p-NSS-CLT}
The number $224p$ is an NSS-CLT order, for all primes $p \ne 2, 3, 5, 7, 31$.

\end{theorem}

\begin{corollary}\label{cor:inf-NSS-CLT}
There are infinitely many NSS-CLT orders.
\end{corollary}

\section{Two conjectures and a characterization theorem}

Table~\ref{tab:NSS-NCLT}, generated by our computer program, includes the 12
NSS-CLT orders up to 10,000.  We performed an extended run of this program to
produce a list of all NSS-CLT orders up to 1,000,000,000.  Inspecting the
resulting list gave rise to a pair of conjectures, described below, along with
a remarkable characterization of the most common form of NSS-CLT order.

We first found the percentage of integers, less than or equal to a given limit,
which are NSS-CLT orders.  Letting
\begin{center}
$f_1( n )$ \,=\, the number of NSS-CLT orders up to $n$
\end{center}
we obtain the following:

\begin{center}
\begin{tabular}{rcrcr}
$n$ & & $f_1(n)$ & & $f_1(n) / n$          \\
\cline{1-1} \cline{3-3} \cline{5-5}        \\[-1.8ex]
       10,000 & &        12 & & 0.120000\% \\
      100,000 & &       107 & & 0.107000\% \\
    1,000,000 & &     1,094 & & 0.109400\% \\
   10,000,000 & &    10,889 & & 0.108890\% \\
  100,000,000 & &   108,906 & & 0.108906\% \\
1,000,000,000 & & 1,089,086 & & 0.108909\% \\
\end{tabular}
\captionof{table}{Number of NSS-CLT Orders}
\label{tab:percent-NSS-CLT}
\end{center}

This leads us to propose

\begin{conjecture}\label{cnj:percent-NSS-CLT}
The proportion of positive integers which are NSS-CLT orders converges to a
non-zero constant, whose value is approximately 0.1089\%.
\end{conjecture}

\noindent
It is a little surprising to us that a \emph{fixed} proportion of orders appear
to be NSS-CLT.

Looking back at Table~\ref{tab:NSS-NCLT}, observe that no fewer than 10 of its
12 NSS-CLT orders are multiples\footnote{We consider 224 itself to be a
\emph{multiple} of 224.} of 224, the smallest such order.  We investigated this
further.  Letting
\begin{center}
$f_2( n )$ \,=\, the number of NSS-CLT orders up to $n$ which are multiples of
224
\end{center}
we extend Table~\ref{tab:percent-NSS-CLT} to obtain the following:

\begin{center}
\begin{tabular}{rcrcrcr}
$n$ & & $f_2(n)$ & & $f_1(n)$ & & $f_2(n) / f_1(n)$   \\
\cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7}       \\[-1.8ex]
       10,000 & &      10 & &        12 & &  83.333\% \\
      100,000 & &      95 & &       107 & &  88.785\% \\
    1,000,000 & &     961 & &     1,094 & &  87.843\% \\
   10,000,000 & &   9,584 & &    10,889 & &  88.015\% \\
  100,000,000 & &  95,846 & &   108,906 & &  88.008\% \\
1,000,000,000 & & 958,550 & & 1,089,086 & &  88.014\% \\
\end{tabular}
\captionof{table}{Number of NSS-CLT Orders which are multiples of 224}
\label{tab:percent-224-mult}
\end{center}

\begin{conjecture}\label{cnj:percent-224-mult}
The proportion of NSS-CLT orders which are multiples of 224 converges to a
non-zero constant, whose value is approximately 88\%.
\end{conjecture}

\noindent
Again, it is a little surprising to us that a \emph{fixed} proportion of NSS-CLT
orders --- and a large one at that --- appear to be multiples of 224.  Of
course, from Theorem~\ref{thm:224p-NSS-CLT} we already know that $224p$ is an
NSS-CLT order, for all primes $p \ne 2, 3, 5, 7, 31$.

Although the values of $n$ chosen for Table~\ref{tab:percent-NSS-CLT} and
Table~\ref{tab:percent-224-mult} are all consecutive powers of 10, we also
generated data for several other random intermediate values of $n$, and the
results are entirely consistent with the above tables.

Due to their relative abundance, we then looked more closely at those NSS-CLT
orders which are proper multiples of 224.  Besides those of the form $224p$ (see
Theorem~\ref{thm:224p-NSS-CLT}), we have

\begin{center}
{\footnotesize{%
  \begin{tabular}{rclcc}
    \LINE{32032} {224 \D 11 \D 13} {}
    \LINE{41888} {224 \D 11 \D 17} {}
    \LINE{46816} {224 \D 11 \D 19} {}
    \LINE{49504} {224 \D 13 \D 17} {}
    \LINE{55328} {224 \D 13 \D 19} {}
    \LINE{56672} {224 \D 11 \D 23} {}
    \LINE{66976} {224 \D 13 \D 23} {}
    \LINE{71456} {224 \D 11 \D 29} {}
    \LINE{72352} {224 \D 17 \D 19} {}
    \LINE{84448} {224 \D 13 \D 29} {}
    \LINE{87584} {224 \D 17 \D 23} {}
  \end{tabular}
  \hspace{-1.0em}
  \begin{tabular}{rclcc}
    \LINE{  91168} {224 \D 11 \D 37}       {}
    \LINE{  97888} {224 \D 19 \D 23}       {}
    \LINE{ 101024} {224 \D 11 \D 41}       {}
    \GAP
    \LINE{ 544544} {224 \D 11 \D 13 \D 17} {}
    \LINE{ 608608} {224 \D 11 \D 13 \D 19} {}
    \LINE{ 736736} {224 \D 11 \D 13 \D 23} {}
    \LINE{ 795872} {224 \D 11 \D 17 \D 19} {}
    \LINE{ 928928} {224 \D 11 \D 13 \D 29} {}
    \LINE{ 940576} {224 \D 13 \D 17 \D 19} {}
    \LINE{ 963424} {224 \D 11 \D 17 \D 23} {}
  \end{tabular}
  \hspace{-1.0em}
  \begin{tabular}{rclcc}
    \LINE{ 1076768} {224 \D 11 \D 19 \D 23}       {}
    \LINE{ 1138592} {224 \D 13 \D 17 \D 23}       {}
    \LINE{ 1185184} {224 \D 11 \D 13 \D 37}       {}
    \LINE{ 1214752} {224 \D 11 \D 17 \D 29}       {}
    \LINE{ 1272544} {224 \D 13 \D 19 \D 23}       {}
    \LINE{ 1313312} {224 \D 11 \D 13 \D 41}       {}
    \GAP
    \LINE{10346336} {224 \D 11 \D 13 \D 17 \D 19} {}
    \LINE{12524512} {224 \D 11 \D 13 \D 17 \D 23} {}
    \LINE{13997984} {224 \D 11 \D 13 \D 19 \D 23} {}
    \GAP
  \end{tabular}}}
\captionof{table}{NSS-CLT non-prime multiples of 224}
\label{tab:224-non-prime-mult}
\end{center}

The pattern emerging here is surprising and really quite remarkable.  For ease
of reference, we temporarily introduce the following phrase:

\begin{definition}
A \emph{special number} is any integer of the form
\EXPR{224\,p_1\,p_2\,\cdots\,p_k}
where $k \ge 0$, each $p_i$ is a distinct prime, and no $p_i$ equals 2, 3, 5, 7,
or 31.
\end{definition}

\begin{theorem}\label{thm:224-mult-special}
The NSS-CLT multiples of 224 are precisely the special numbers.
\end{theorem}

\begin{proof}
Note at the outset that, by Theorem~\ref{thm:NSSmult}, every multiple of the NSS
order 224 is itself an NSS order.  So we need only show that the CLT multiples
of 224 are the special numbers.

Our proof uses both the characterization of CLT orders given in
Berger~\cite{Be}, and the notation of that paper.

We first show that every CLT multiple of 224 is a special number.  Accordingly,
let $n$ be any CLT multiple of 224 ($= 2^5\,7$) and let
\EXPR{2^5\,7\,p_1\,p_2\,\cdots\,p_k}
be its prime factorization, for some $k \ge 0$.  We show that the primes $p_i$
are pairwise distinct, and that no $p_i$ equals 2, 3, 5, 7, or 31.

Let $q$ be any odd prime, and express $n$ as \,$n = \ell\,2^a q^b$, where
$2 \nmid \ell$ and $q \nmid \ell$ (so $a$ and $b$ are the highest powers of $2$
and $q$, respectively, in $n$).  Since $a \ge 5$, Proposition~3.6~\cite{Be}
gives $b = 0$ or $b = 1$.  Thus the primes $p_i$ are pairwise distinct.
Moreover, no $p_i$ equals 7.

Now express $n$ as \,$n = \ell\,7^1 2^b$, where $7 \nmid \ell$ and
$2 \nmid \ell$.  Since the exponent of 2 (mod 7) is~3, Proposition~3.5~\cite{Be}
gives \,$b \in \{0, 1, 2, 5\}$.  In particular, $b \le 5$, so no $p_i$ can equal
2.

Finally, express $n$ as \,$n = \ell\,p\,2^b$, where $b = 5$, $p$ is an odd
prime, $p \nmid \ell$, and $2 \nmid \ell$.  Letting $d$ denote the exponent of
2 (mod $p$), consider the possibilities:
\\[0.5ex]
\CASEa{3}{2}{\{ 0, 1 \}}
\\
\CASEa{5}{4}{\{ 0, 1, 2, 3 \}}
\\
\CASEa{31}{5}{\{ 0, 1, 2, 3, 4, 9 \}}
\\[0.5ex]
and by Proposition~3.5~\cite{Be}, $n$ cannot be a CLT order in any of these
three cases.  Thus, no $p_i$ equals 3, 5, or 31, completing the proof that every
CLT multiple of 224 is a special number.

We now show that every special number, clearly being a multiple of 224, is a CLT
order, by showing that it is ``good''~\cite{Be}.  Accordingly, consider any
special number
\EXPR{n \,=\, 2^5\,7\,p_1\,p_2\,\cdots\,p_k \,=\, \prod_{q \in Q}\,q^{e(q)}}
where\,
$Q = \{2, 7, p_1, \cdots, p_k\},\, e(2) = 5,\, e(7) = 1,\,$ and $e(p_i) = 1$ for
$1 \le i \le k$.  We first show that $e(q) \in \mathscr{S}(m, q)$ for each
divisor $m$ of $n$ and each $q \in Q$ (Berger~\cite{Be} defines the set
$\mathscr{S}$\,).

If $m$ is composite, then \,$\mathscr{S}(m, q) = \POSITIVES$\, for each $q$, so
$e(q) \in \mathscr{S}(m, q)$.  Otherwise, $m$ is a prime divisor
of $n$, so in fact $m \in Q$.  For $q = 2$, we have the following cases; again,
Berger~\cite{Be} defines the sets $\mathscr{I}$ and $\mathscr{I}'$:
\\[0.5ex]
\CASEb{2}{\POSITIVES}
\\
\CASEb{7}{\mathscr{I}(3) \,=\, \{1, 2, 5\}}
\\
\CASEb{p_i}{\mathscr{I}(d)} or $\mathscr{I}'(d),\, d \ge 6$ \,(check
                                         $p_i = 11, 13, 17, 19, 23, 29$, and
                                         $p_i \ge 37$)
\\[0.5ex]
and in each case, $e(2) = 5 \in \mathscr{S}(m,2)$.  For all other cases
($q \in Q, q \ne 2$), we have $e(q) = 1$, and since 1 always belongs to the set
$\mathscr{S}(m,q)$, this gives $e(q) \in \mathscr{S}(m,q)$.

We next show that $e(q) \in \mathscr{S}(r, p^u, q)$ for each prime $r$ and $p$
and positive integer $u$ such that $rp^u$ divides $n$, and each $q \in Q$.

For $q = 2$, there are no primes $r$ and $p$ for which $rp \mid q-1$,
giving $\mathscr{S}(r, p^u, q) = \POSITIVES$ and thus
$e(q) = 5 \in \mathscr{S}(r, p^u, q)$.  For all other cases
($q \in Q, q \ne 2$), we again have $e(q) = 1$, and since 1 always belongs to
the set $\mathscr{S}(r, p^u, q)$, this gives $e(q) \in \mathscr{S}(r, p^u, q)$.

So the special number $n$ is ``good'', and by Theorem~1.1~\cite{Be}, is a CLT
order.
\end{proof}

Theorem~\ref{thm:224-mult-special} greatly extends
Theorem~\ref{thm:224p-NSS-CLT}; nonetheless, we opted to retain the earlier
result in this paper due to its significantly different and simpler proof.

We conclude by noting, with appreciation, the immense contribution of the
computer to~the above work.  To \emph{manually} determine the NSS-CLT status of
even a single integer, using the results of Berger and Pazderski\,/\,Hughes,
would be both tedious and error-prone.  Yet in under 16 hours on a modest
desktop, our program had done so for the first billion positive integers.  The
resulting list of NSS-CLT orders was useful for several of our theorems, and
essential for both of our conjectures.  In particular, the surprising pattern of
Theorem~\ref{thm:224-mult-special} would not have been apparent without such a
very extensive list of NSS-CLT orders.  Thus, in these explorations, as in
several other areas of discrete mathematics, the computer proves to be a
valuable tool and a most helpful assistant.

\section{Acknowledgment}

Our sincere thanks to an anonymous referee for guiding us to the proof of
Theorem~\ref{thm:224-mult-special}.

\begin{thebibliography}{99}

\bibitem{Be}
T. R. Berger, A converse to Lagrange's theorem, \emph{J.\ Aust.\ Math.\ Soc.}
(Ser.\ A) \textbf{25} (1978), 291--313.

\bibitem{BM}
M. Brennan and D. MacHale, Variations on a theme: $A_4$ definitely
has no subgroup of order six!, \emph{Math.\ Mag.} \textbf{73} (2000), 36--40.

\bibitem{Br}
H. G. Bray, A note on CLT groups, \emph{Pacific J.\ Math.} \textbf{27} (1968),
229--231.

\bibitem{Bu}
W. Burnside, \emph{Theory of Groups of Finite Order}, Cambridge University
Press, 1897.  Available online at \url{http://www.gutenberg.org/ebooks/40395}\,.

\bibitem{Cu}
M. J. Curran, Non-CLT groups of small order, \emph{Comm.\ Algebra} \textbf{11}
(1983), 111--126.

\bibitem{De}
W. E. Deskins, A characterization of finite supersolvable groups,
\emph{Amer.\ Math.\ Monthly} \textbf{75} (1968), 180--182.

\bibitem{Ha}
P. Hall, A note on soluble groups, \emph{J.\ London Math.\ Soc.\ (2)}
\textbf{1-3} (1928), 98--105.

\bibitem{Hu}
A. Hughes, Automorphisms of nilpotent groups and supersolvable orders,
\emph{Proc.\ Sympos.\ Pure Math.} \textbf{37} (1980), 205--207.

\bibitem{HJ}
J. F. Humphreys and D. L. Johnson, On Lagrangian groups,
\emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{180} (1973), 291--300.

\bibitem{Pa}
G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft
geh\"{o}ren, \emph{Arch.\ Math.} \textbf{10} (1959), 331--343.

\bibitem{Sl}
N. J. A. Sloane, \emph{The On-Line Encyclopedia of Integer Sequences},
\url{http://oeis.org}\,.

\bibitem{St}
R. R. Struik, Partial converses to Lagrange's theorem, \emph{Comm.\ Algebra}
\textbf{6} (1978), 421--482.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent
2010 \emph{Mathematics Subject Classification}:
Primary 20F16; Secondary 20D20, 20K27, 68R05.

\noindent
\emph{Keywords}:
finite group, converse Lagrange theorem order, supersolvable order.

\bigskip
\hrule
\bigskip

\noindent
(Concerned with sequence \seqnum{A066085}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 24 2015;
revised version received November 4 2016.
Published in {\it Journal of Integer Sequences}, November 13 2016.
Slight revision of Tables 2 and 3, June 20 2017.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\end{document}
