Janko group

Group theory

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko group J1, 2, 3, 4 Fischer group F22..24 Baby Monster group B Monster group M Other finite groups Symmetric group, Sn Dihedral group, Dn Infinite groups The integers, Z Modular groups, PSL(2,Z) and SL(2,Z) Topological groups Lie 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 Infinite dimensional groups conformal group diffeomorphism group Loop group Quantum group O(∞) SU(∞) Sp(∞)

This box: view • talk • edit

In mathematics, a Janko group is one of the four sporadic simple groups named for Zvonimir Janko. Janko constructed the first Janko group J₁ in 1965. At the same time, Janko also predicted the existence of J₂ and J₃. In 1976, he suggested the existence of J₄. J₂, J₃, and J₄ were all later shown to exist.

Janko groups

• The Janko group J1 has order 175 560 = 2³ · 3 · 5 · 7 · 11 · 19. It is the only Janko group whose existence was proved by Janko himself.
• The Hall–Janko group has order 604 800 = 2⁷ · 3³ · 5² · 7. It is also known as J₂, HJ, or the Hall–Janko–Wales group. It was constructed by Marshall Hall, Jr., and David Wales.
• The Janko group J3 has order 50 232 960 = 2⁷ · 3⁵ · 5 · 7 · 19. It is also known as the Higman–Janko–McKay group. It was constructed by Graham Higman and John McKay.
• The Janko group J4 has order 86 775 571 046 077 562 880 = 2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43. It was constructed by Simon Norton.

This article includes a list of related items that share the same name (or similar names).