Classification of finite simple groups

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A vast body of work, mostly published between around 1955 and 1983, classifies all of the finite simple groups. The classification shows all finite simple groups to be one of the following types:

  • a cyclic group with prime order
  • an alternating group of degree at least 5
  • a classical linear group
  • an exceptional or twisted group of Lie type (including the Tits group)
  • or one of 26 left-over groups known as the sporadic groups

The Sporadic Groups

5 of the sporadic groups were discovered by Mathieu in the 1860's and the other 21 were found between 1965 and 1975. The full list is:

  • Mathieu groups M11, M12, M22, M23, M24
  • Janko groups J1, J2, J3, J4
  • Conway groups Co1,Co2,Co2
  • Fischer groups F22,F23,F24
  • Higman-Sims group HS
  • McLaughlin group McL
  • Held group He
  • Rudvalis group Ru
  • Suzuki sporadic group Suz
  • O'Nan group ON
  • Harada-Norton group HN
  • Lyons group Ly
  • Thompson group Th
  • Baby Monster group
  • Monster group M

The Monster Group

The largest of the sporadic groups is the Monster group. It has 246.320.59.76.112.13317.19.23.29.31.41.47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements and plays a starring role in the Monstrous Moonshine Conjectures of Conway and Morton which were subsequently proved by Borcherds.

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