3 deleted 4 characters in body

I think the claim goes back to the 1979 paper "Monstrous Moonshine" by Conway and Norton, where they discuss the "defining property of 24": If $n$ is a positive integer such that $xy \equiv 1$ mod $n$ implies $x \equiv y$, then $n|24$. This fact is used in Atkin's determination of the normalizer of $\Gamma_0(N)$ in $SL_2(\mathbb{R})$. The number 24 plays a special role here, in the sense that the normalizer is $\Gamma_0(n|h)+$, where:

1. $h$ is the largest divisor of 24 such that $h^2|N$
2. $n = N/h$
3. $\Gamma_0(n|h) = \left\{ \begin{pmatrix} a & b/h \\ cN & d \end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad-bcn/h = 1 \right\}$. The notation is meant to suggest that the group is conjugate to $\Gamma_0(n/h)$, but contained in and contains $\Gamma_0(nh)$.
4. The $+$ means we adjoin all possible Atkin-Lehner involutions. If $n/h$ has $k$ prime factors, then this extends $\Gamma_0(n|h)$ by an elementary abelian 2-group of rank $k$.

The connection between your observation and moonshine does not seem particularly strong to me, but that may be because I was too young to have experienced firsthand the heady days of numerical exploration. It involves the normalizers of $\Gamma_0(N)$ in the following way: There is a graded representation $V^\natural = \bigoplus V^\natural_m$ of the monster, such that for each element $g$ of the monster, the McKay-Thompson series $T_g(\tau) = \sum_{m \geq -1} Tr(g|V^\natural_m)q^m$ is the $q$-expansion of a modular function invariant under some genus zero group $\Gamma$ that contains and normalizes some $\Gamma_0(N)$, and therefore lies in some $\Gamma_0(n|h)+$ , where $n = |g|$. This fact was essentially the main conjecture in the Conway-Norton paper, although the paper enhances this claim with an explicit list of the candidate functions and their invariance groups. My understanding of the solution process is:

1. Atkin, Fong, and Smith gave a computational proof of existence (1980).
2. Frenkel, Lepowsky, and Meurman constructed a candidate representation $V^\natural$ (1984), and showed that it had a vertex operator algebra structure (1988).
3. Borcherds proved that the candidate representation was satisfactory (1992).

Wikipedia and sundry expository books by Gannon, du Sautoy, Ronan, and others can say more about the precise history than I can. I should mention that the number 24 is important as the central charge of $V^\natural$ in Borcherds's solution to the Monstrous Moonshine conjecture, but the paper does not make explicit use of the number 24 in the "group of units" role. One might reasonably argue (through a somewhat convoluted path) that these are the same 24, though.

There may be other connections, but I am unaware of them.

2 Fixed some errors, made lists.

I think the claim goes back to the 1979 paper "Monstrous Moonshine" by Conway and Norton, where they discuss the "defining property of 24": If $n$ is a positive integer such that $xy \equiv 1$ mod $n$ implies $x \equiv y$, then $n|24$. This fact is used in Atkin's determination of the normalizer of $\Gamma_0(N)$ in $SL_2(\mathbb{R})$. The number 24 plays a special role here, in the sense that the normalizer is $\Gamma_0(n|h)+$, where:

• $h$ is the largest divisor of 24 such that $h^2|N$, h^2|N$•$n = N/h$, and • $\Gamma_0(n|h) = \left\{ \begin{pmatrix} a & b/h \\ cN & d \end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad-bcn/h = 1 \right\}$. The notation is meant to suggest that the group is conjugate to$\Gamma_0(n/h)$, but contained in$\Gamma_0(nh)$. • The$+$means we adjoin the all possible Atkin-Lehner involutions. If$w_s$corresponding to all n/h$ has $s$ such that k$prime factors, then this extends$s|N$and \Gamma_0(n|h)$ by an elementary abelian 2-group of rank $(s,N/s)=1$.k$. • The connection between your observation and moonshine does not seem particularly strong to me, but it that may be because I was too young to have experienced firsthand the heady days of numerical exploration. It involves the normalizers of$\Gamma_0(N)$in the following way: There is a graded representation$V^\natural = \bigoplus V^\natural_m$of the monster, such that for each element$g$of the monster, the McKay-Thompson series$T_g(\tau) = \sum_{m \geq -1} Tr(g|V^\natural_m)q^m$is the$q$-expansion of a modular function invariant under some genus zero group$\Gamma$that contains and normalizes some$\Gamma_0(N)$, and therefore lies in some$\Gamma_0(n|h)+$, where$n = |g|$. This fact was essentially the main conjecture in the Conway-Norton paper, although the paper makes a stronger enhances this claim involving with an explicit list of the candidate functions and their invariance groups. My understanding of the solution process is: • Atkin, Fong, and Smith gave a computational proof of existence , (1980). • Frenkel, Lepowsky, and Meurman constructed a candidate representation ,$V^\natural$(1984), and showed that it had a vertex operator algebra structure (1988). • Borcherds proved that the candidate representation was satisfactory (1992). • Wikipedia and sundry expository books by Gannon, du Sautoy, Ronan, and others can say more about the precise history than I can. I should mention that the number 24 is important as the central charge of$V^\natural$in Borcherds's solution to the Monstrous Moonshine conjecture, but the paper does not make explicit use of the number 24 in the "group of units" role. One might reasonably argue (through a somewhat convoluted path) that these are the same 24, though. 1 I think the claim goes back to the 1979 paper "Monstrous Moonshine" by Conway and Norton, where they discuss the "defining property of 24": If$n$is a positive integer such that$xy \equiv 1$mod$n$implies$x \equiv y$, then$n|24$. This fact is used in Atkin's determination of the normalizer of$\Gamma_0(N)$in$SL_2(\mathbb{R})$. The number 24 plays a special role here, in the sense that the normalizer is$\Gamma_0(n|h)+$, where$h$is the largest divisor of 24 such that$h^2|N$,$n = N/h$, and the$+$means we adjoin the Atkin-Lehner involutions$w_s$corresponding to all$s$such that$s|N$and$(s,N/s)=1$. The connection between your observation and moonshine does not seem particularly strong to me, but it involves the normalizers of$\Gamma_0(N)$in the following way: There is a graded representation$V^\natural = \bigoplus V^\natural_m$of the monster, such that for each element$g$of the monster, the McKay-Thompson series$T_g(\tau) = \sum_{m \geq -1} Tr(g|V^\natural_m)q^m$is the$q$-expansion of a modular function invariant under some genus zero group$\Gamma$that contains and normalizes some$\Gamma_0(N)$, and therefore lies in some$\Gamma_0(n|h)+$, where$n = |g|\$. This fact was the main conjecture in the Conway-Norton paper, although the paper makes a stronger claim involving an explicit list of the candidate functions. Atkin, Fong, and Smith gave a computational proof of existence, Frenkel, Lepowsky, and Meurman constructed a candidate representation, and Borcherds proved that the candidate representation was satisfactory.

There may be other connections, but I am unaware of them.