Fischer group

Group theory

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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. They constitute three of the 26 sporadic groups. Sometimes the term encompasses their automorphism groups.

3-transposition groups

The Fischer groups are 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 no uniformly accepted notation for these groups. Some authors use F in place of Fi (F₂₂, for example). 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

• 3-Transposition Groups (Cambridge Tracts in Mathematics) by Michael Aschbacher. "3-Transposition Groups" contains the first published proof of Fischer's Theorem, written out completely in one place. Publisher: Cambridge University Press (November 28, 1996) ISBN 0-521-57196-0