View source for Harada–Norton group

{{Short description|A sporadic simple group}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[Group theory|group theory]], the '''Harada–Norton group''' ''HN'' is a sporadic simple group of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>14</sup>{{·}}3<sup>6</sup>{{·}}5<sup>6</sup>{{·}}7{{·}}11{{·}}19
: = 273030912000000
: ≈ 3{{e|14}}.

==History and properties==
''HN'' is one of the 26 sporadic groups and was found by {{harvs|txt |last=Harada |year=1976}} and {{harvs|txt |last=Norton |authorlink=Simon P. Norton |year=1975}}).

Its [[Schur multiplier]] is trivial and its [[Outer automorphism group|outer automorphism group]] has order 2. 

''HN'' has an involution whose [[Centralizer and normalizer|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|vertex operator algebra]] over the field with 5 elements {{harv|Lux|Noeske|Ryba|2008}}. This implies that it acts on a 133 dimensional algebra over '''F'''<sub>5</sub> with a commutative but nonassociative product, analogous to the [[Griess algebra]] {{harv|Ryba|1996}}.

==Generalized monstrous moonshine==

Conway and Norton suggested in their 1979 paper that [[Monstrous moonshine|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. 
To recall, the prime number 5 plays a special role in the group and for ''HN'', the relevant McKay-Thompson series is <math>T_{5A}(\tau)</math> where one can set the constant term {{nowrap|a(0) {{=}} −6}} ({{OEIS2C|id=A007251}}),

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

and ''η''(''τ'') is the [[Dedekind eta function]].

==Maximal subgroups==
{{harvtxt|Norton|Wilson|1986}} found the 14 conjugacy classes of [[Maximal subgroup|maximal subgroup]]s of ''HN'' as follows:

* A<SUB>12</SUB>
* 2.HS.2
* U<SUB>3</SUB>(8):3
* 2<SUP>1+8</SUP>.(A<SUB>5</SUB> × A<SUB>5</SUB>).2
* (D<SUB>10</SUB> × U<SUB>3</SUB>(5)).2
* 5<SUP>1+4</SUP>.2<SUP>1+4</SUP>.5.4
* 2<SUP>6</SUP>.U<SUB>4</SUB>(2)
* (A<SUB>6</SUB> × A<SUB>6</SUB>).D<SUB>8</SUB>
* 2<SUP>3+2+6</SUP>.(3 × L<SUB>3</SUB>(2))
* 5<SUP>2+1+2</SUP>.4.A<SUB>5</SUB>
* M<SUB>12</SUB>:2 (Two classes, fused by an outer automorphism)
* 3<SUP>4</SUP>:2.(A<SUB>4</SUB> × A<SUB>4</SUB>).4
* 3<SUP>1+4</SUP>:4.A<SUB>5</SUB>

==References==
*{{Citation | last1=Harada | first1=Koichiro | title=Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) | publisher=[[Company:Academic Press|Academic Press]] | location=Boston, MA | mr=0401904 | year=1976 | chapter=On the simple group ''F'' of order&nbsp;2<sup>14</sup>&nbsp;·&nbsp;3<sup>6</sup>&nbsp;·&nbsp;5<sup>6</sup>&nbsp;·&nbsp;7&nbsp;·&nbsp;11&nbsp;·&nbsp;19 | pages=119–276}}
*{{Citation | last1=Lux | first1=Klaus | last2=Noeske | first2=Felix | last3=Ryba | first3=Alexander J. E. | title=The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 | doi=10.1016/j.jalgebra.2007.03.046 | mr=2378074 | year=2008 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=319 | issue=1 | pages=320–335| doi-access=free }}
*S. P. Norton, ''F and other simple groups'', PhD Thesis, Cambridge 1975.
*{{Citation | last1=Norton | first1=S. P. | last2=Wilson | first2=Robert A. | title=Maximal subgroups of the Harada-Norton group | doi=10.1016/0021-8693(86)90192-4 | mr=860712 | year=1986 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=103 | issue=1 | pages=362–376| doi-access=free }}
*{{Citation | last1=Ryba | first1=Alexander J. E. | title=A natural invariant algebra for the Harada-Norton group | doi=10.1017/S0305004100074454 | mr=1362942 | year=1996 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=119 | issue=4 | pages=597–614}}

==External links==
* [http://mathworld.wolfram.com/Harada-NortonGroup.html MathWorld: Harada–Norton Group]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HN/ Atlas of Finite Group Representations: Harada–Norton group]

{{DEFAULTSORT:Harada-Norton group}}
[[Category:Sporadic groups]]

{{Sourceattribution|Harada–Norton group}}