Sporadic Group - Names of The Sporadic Groups

Names of The Sporadic Groups

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:

• Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
• Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
• Conway groups Co₁ or F₂₋, Co₂, Co₃
• Fischer groups Fi₂₂, Fi₂₃, Fi₂₄′ or F₃₊
• Higman–Sims group HS
• McLaughlin group McL
• Held group He or F₇₊ or F₇
• Rudvalis group Ru
• Suzuki sporadic group Suz or F₃₋
• O'Nan group O'N
• Harada–Norton group HN or F₅₊ or F₅
• Lyons group Ly
• Tits group T
• Thompson group Th or F_3|3 or F₃
• Baby Monster group B or F₂₊ or F₂
• Fischer–Griess Monster group M or F₁

Matrix representations over finite fields for all the sporadic groups have been constructed.

The earliest use of the term "sporadic group" may be Burnside (1911, p. 504, note N) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received".

Diagram is based on diagram given in Ronan (2006). The sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous.