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the 171 moonshine groups

Monstrous moonshine associates to every element of order n of the monster group M an arithmetic group of the form
(n|h)+e,f,
where h is a divisor of 24 and of n and where e,f, are divisors of nh coprime with its quotient.

In snakes, spines, and all that we’ve constructed the arithmetic group
Γ0(n|h)+e,f,
which normalizes Γ0(N) for N=h.n. If h=1 then this group is the moonshine group (n|h)+e,f,, but for h>1 the moonshine group is a specific subgroup of index h in Γ0(n|h)+e,f,.

I’m sure one can describe this subgroup explicitly in each case by analysing the action of the finite group (Γ0(n|h)+e,f,)/Γ0(N) on the (N|1)-snake. Some examples were worked out by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.

But at the moment I don’t understand the general construction given by Conway, McKay and Sebbar in On the discrete groups of moonshine. I’m stuck at the last sentence of (2) in section 3. Nothing a copy of Charles Ferenbaugh Ph. D. thesis cannot fix.

The correspondence between the conjugacy classes of the Monster and these arithmetic groups takes up 3 pages in Conway & Norton’s Monstrous Moonshine. Here’s the beginning of it.

Published in groups math