Monstrous Moonshine Wikipedia claims that the group of units of Z24 (1,5,7,11,13,17,19,23), which all have order 2, and are isomorphic to (Z/2Z)^3 have an important connection to Monstrous Moonshine theory, however, I cannot find any other reference besides Wikipedia that claims this --- It was recommended on sci.math that I pose this question here.
Perhaps it's a mistake? And he meant that the primes of the Monster, which continue to 71, are what are considered in Moonshine.
Paul Hjelmstad, B.M, B.A.
[Edit (PLC): Here is the relevant passage from wikipedia:]

24 is the highest number $n$ with the property that every element of the group of units $(\mathbb{Z}/n\mathbb{Z})^{\times}$ of the commutative ring $\mathbb{Z}/n\mathbb{Z}$, apart from the identity element, has order $2$; thus the multiplicative group $(\mathbb{Z}/24\mathbb{Z})^{\times} = \{1,5,7,11,13,17,19,23\}$ is isomorphic to the additive group $(\mathbb{Z}/2\mathbb{Z})^3$. This fact plays a role in monstrous moonshine.

 A: $\newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$
I don't know about monstrous moonshine, but $(\Z/24\Z)^\times$ is the group of automorphisms of the maximal elementary abelian $2$-extension $\Q_2\left(\root2\of{\Q_2^\times}\right)=\Q_2(\root2\of5, \root2\of3, \root2\of2)=\Q_2(\zeta_{24})$ of $\Q_2$.  See for example Lemma 8 of Lecture 19 of my course  on Local arithmetic.
A: 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 $h^2|N$

*$n = N/h$

*$\Gamma_0(n|h) = \left\{ \begin{pmatrix} a & b/h \\ cN & d \end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad-bcn/h = 1 \right\}$.  The notation is meant to suggest that the group is conjugate to $\Gamma_0(n/h)$, and contains $\Gamma_0(nh)$.

*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:


*

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