In the mathematical field of group theory, the Harada–Norton group HN, found by Template:Harvs and Template:Harvs) is a sporadic simple group of order

214Template:·36Template:·56Template:·7Template:·11Template:·19
= 273030912000000
≈ 3Template:·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 Template:Harv. This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the Griess algebra Template:Harv.

Generalized Monstrous Moonshine

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 ${\displaystyle T_{5A}(\tau )}$ where one can set the constant term a(0) = -6 (),

{\displaystyle {\begin{aligned}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{aligned}}}

and η(τ) is the Dedekind eta function.

Maximal subgroups

Template:Harvtxt 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

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• S. P. Norton, F and other simple groups, PhD Thesis, Cambridge 1975.
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