Harada–Norton group

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In the mathematical field of group theory, the Harada–Norton group HN, found by Harada (1976) and Norton (1975)) is a sporadic simple group of order

   214 · 36 · 56 · 7 · 11 · 19
= 273030912000000
≈ 3 · 1014.

Its Schur multiplier is trivial and its outer automorphism group has order 2.

The Harada–Norton group has an involution whose centralizer is of the form 2.HS.2, where HS is the Higman-Sims group (which is how Harada found it).

The prime 5 plays a special role in the group. For example, it centralizes an element of order 5 in the Monster group (which is how Norton found it), and as a result acts naturally on a vertex operator algebra over the field with 5 elements (Lux, Noeske & Ryba 2008). This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the Griess algebra (Ryba 1996).

Generalized Monstrous Moonshine[edit]

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For HN, the relevant McKay-Thompson series is T_{5A}(\tau) where one can set the constant term a(0) = -6 (OEISA007251),

\begin{align}j_{5A}(\tau)
&=T_{5A}(\tau)-6\\
&=\big(\tfrac{\eta(\tau)}{\eta(5\tau)}\big)^{6}+5^3 \big(\tfrac{\eta(5\tau)}{\eta(\tau)}\big)^{6}\\
&=\frac{1}{q} - 6 + 134q + 760q^2 +3345q^3+12256q^4+39350q^5+\dots
\end{align}

and η(τ) is the Dedekind eta function. Let r_n = 1, 133, 133, 760, 3344, 8778, 8778, 8910,...(OEISA003915) be the degrees of irreducible representations of HN. Then,

\begin{align}
1 &= r_1\\
134&= r_1+r_2\\
760&= r_4\\
3345 &= r_1+r_5\\
12256&= r_1+r_2+r_5+r_6\\
39350&=1+133+760+3344+35112
\end{align}

and so on. Like the j-function, the function j5A(τ) can also assume an integer value for appropriate arguments. For example,

j_{5A}\Big(\tfrac{5+\sqrt{-235}}{10}\Big) = -15250

Maximal subgroups[edit]

Norton & Wilson (1986) found the 14 classes of maximal subgroups as follows:

A12

2.HS.2

U3(8):3

21+8.(A5 × A5).2

(D10 × U3(5)).2

51+4.21+4.5.4

26.U4(2)

(A6 × A6).D8

23+2+6.(3 × L3(2))

52+1+2.4.A5

M12:2 (Two classes, fused by an outer automorphism)

34:2.(A4 × A4).4

31+4:4.A5

References[edit]

External links[edit]