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these groups. 3-TRANSPOSITION GROUPS The Fischer groups are finite groups named after Bernd Fischer, who discovered them while investigating 3-transposition groups. These are groups ''G'' with the following properties:
The typical example of a 3-transposition group is a Symmetric Group , where the Fischer transpositions are genuinely transpositions. Fischer was able to classify 3-transposition groups which satisfy certain extra technical conditions. The groups he found fell into several infinite classes (as well as the symmetric groups, certain classes of symplectic and orthogonal groups fulfilled his conditions) with the exception of the three Fischer groups. These groups are usually referred to as ''Fi''22, ''Fi''23 and ''Fi''24. The first two of these are simple groups, and the third contains the simple group ''Fi''24' of index 2. ORDERS The ''order'' of a group is the number of elements in the group. ''Fi''22 has order 217.39.52.7.11.13 = 64561751654400. ''Fi''23 has order 218.313.52.7.11.13.17.23 = 4089470473293004800. ''Fi''24' has order 221.316.52.73.11.13.17.23.29 = 1255205709190661721292800. It is the 3rd largest of the sporadic groups (after the Monster Group and Baby Monster Group ). NOTATION There is unfortunately no uniformly accepted notation for these groups. Some authors use ''F'' in place of ''Fi'' (e.g. ''F''22). Fischer's notation for the them was ''M''(22), ''M''(23) and ''M''(24)', which emphasised their close relationship with the three largest Mathieu Group s, ''M''22, ''M''23 and ''M''24. One particular source of confusion is that ''Fi''24 is sometimes used to refer to the simple group ''Fi''24', and is sometimes used to refer to the full 3-transposition group (which is twice the size). REFERENCES
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