Harada–Norton group

Group theory
Group theory
Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product
Finite groups (classification) Cyclic group Zn Symmetric group, Sn Dihedral group, Dn Alternating group An Mathieu groups M11, M12, M22, M23, M24 Conway groups Co1, Co2, Co3 Janko groups J1, J2, J3, J4 Fischer groups F22, F23, F24 Baby Monster group B Monster group M
Discrete groups and lattices The integers, Z Lattice (group) Modular groups, PSL(2,Z) and SL(2,Z)
Topological groups and Lie groups Solenoid (mathematics) Circle group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G2 F4 E6 E7 E8 Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

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).

Norton & Wilson (1986) described the maximal subgroups.

References

• Harada, Koichiro (1976), "On the simple group F of order 2¹⁴ · 3⁶ · 5⁶ · 7 · 11 · 19", Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Boston, MA: Academic Press, pp. 119–276, MR0401904 
• Lux, Klaus; Noeske, Felix; Ryba, Alexander J. E. (2008), "The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2", Journal of Algebra 319 (1): 320–335, doi:10.1016/j.jalgebra.2007.03.046, ISSN 0021-8693, MR2378074 
• S. P. Norton, F and other simple groups, PhD Thesis, Cambridge 1975.
• Norton, S. P.; Wilson, Robert A. (1986), "Maximal subgroups of the Harada-Norton group", Journal of Algebra 103 (1): 362–376, doi:10.1016/0021-8693(86)90192-4, ISSN 0021-8693, MR860712 
• Ryba, Alexander J. E. (1996), "A natural invariant algebra for the Harada-Norton group", Mathematical Proceedings of the Cambridge Philosophical Society 119 (4): 597–614, doi:10.1017/S0305004100074454, ISSN 0305-0041, MR1362942 

External links

• Atlas of Finite Group Representations: Harada–Norton group

External links

• MathWorld: Harada–Norton Group