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.
Best Answer
$\newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$
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.