Fischer group
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In mathematics, the term Fischer groups usually refers to the three finite groups denoted Fi22, Fi23, and Fi24', all of which are simple groups, and constitute three of the 26 sporadic groups. Sometimes the term encompasses the automorphism groups of these groups.
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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:
- G is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions'
- The product of any two transpositions has order 1, 2 or 3.
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 Fi22, Fi23 and Fi24. The first two of these are simple groups, and the third contains the simple group Fi24' of index 2.
Orders
The order of a group is the number of elements in the group.
Fi22 has order 217.39.52.7.11.13 = 64561751654400.
Fi23 has order 218.313.52.7.11.13.17.23 = 4089470473293004800.
Fi24' 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. F22). Fischer's notation for the them was M(22), M(23) and M(24)', which emphasised their close relationship with the three largest Mathieu groups, M22, M23 and M24.
One particular source of confusion is that Fi24 is sometimes used to refer to the simple group Fi24', and is sometimes used to refer to the full 3-transposition group (which is twice the size).
References
- Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/html/contents.html#spo - (For the factorisations of the orders) Weisstein, Eric W. "Sporadic Group"
http://mathworld.wolfram.com/SporadicGroup.html