Fischer group
Encyclopedia
In mathematics, the Fischer groups are the three sporadic
simple groups
Fi22, Fi23,Fi24' introduced by .
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. The symmetric group Sn can be generated by n-1 transpositions: (12) ,(23), ..., (n-1,n).
Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new 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.
A starting point for the Fischer groups is the unitary group PSU6(2), which could be thought of as a group Fi21 in the series of Fischer groups, of order 9,196,830,720 = 215.36.5.7.11. Actually it is the double cover 2.PSU6(2) that becomes a subgroup of the new group. This is the stabilizer of one vertex in a graph of 3510 (=2.33.5.13). These vertices become identified as conjugate 3-transpositions in the symmetry group Fi22 of the graph.
The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions all commuting with one another has size 22 and is called a basic set. There are 1024 3-transpositions, called anabasic that do not commute with any in the particular basic set. Any one of other 2364, called hexadic, commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system
, whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22.
The next Fischer group comes by regarding 2.Fi22 as a one-point stabilizer for a graph of 31671 (=34.17.23) vertices, and treating these vertices as the 3-transpositions in a group Fi23. The 3-transpositions come in basic sets of 23, 7 of which commute with a given outside 3-transposition.
Next one takes Fi23 and treats it as a one-point stabilizer for a graph of 306936 (=23.33.72.29) vertices to make a group Fi24. The 3-transpositions come in basic sets of 24, 8 of which commute with a given outside 3-transposition. The group Fi24 is not simple, but its derived subgroup has index 2 and is a sporadic simple 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
).
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, 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).
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
simple groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
Fi22, Fi23,Fi24' introduced by .
3-transposition groups
The Fischer groups are named after Bernd FischerBernd Fischer
Bernd Fischer may refer to:* Bernd Fischer , German mathematician.* Bernd Jürgen Fischer, historian and professor of history at Indiana University-Purdue University Fort Wayne....
who discovered them while investigating 3-transposition groups.
These are groups G with the following properties:
- G is generated by a conjugacy classConjugacy classIn mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
of elements of order 2, called 'Fischer transpositions' or 3-transpositions. - The product of any two distinct transpositions has order 2 or 3.
The typical example of a 3-transposition group is a symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
,
where the Fischer transpositions are genuinely transpositions. The symmetric group Sn can be generated by n-1 transpositions: (12) ,(23), ..., (n-1,n).
Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new 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
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2.
A starting point for the Fischer groups is the unitary group PSU6(2), which could be thought of as a group Fi21 in the series of Fischer groups, of order 9,196,830,720 = 215.36.5.7.11. Actually it is the double cover 2.PSU6(2) that becomes a subgroup of the new group. This is the stabilizer of one vertex in a graph of 3510 (=2.33.5.13). These vertices become identified as conjugate 3-transpositions in the symmetry group Fi22 of the graph.
The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions all commuting with one another has size 22 and is called a basic set. There are 1024 3-transpositions, called anabasic that do not commute with any in the particular basic set. Any one of other 2364, called hexadic, commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system
Steiner system
250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
, whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22.
The next Fischer group comes by regarding 2.Fi22 as a one-point stabilizer for a graph of 31671 (=34.17.23) vertices, and treating these vertices as the 3-transpositions in a group Fi23. The 3-transpositions come in basic sets of 23, 7 of which commute with a given outside 3-transposition.
Next one takes Fi23 and treats it as a one-point stabilizer for a graph of 306936 (=23.33.72.29) vertices to make a group Fi24. The 3-transpositions come in basic sets of 24, 8 of which commute with a given outside 3-transposition. The group Fi24 is not simple, but its derived subgroup has index 2 and is a sporadic simple group.
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
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
and Baby Monster group
Baby Monster group
In the mathematical field of group theory, the Baby Monster group B is a group of orderThe Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the 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 group
Mathieu group
In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
s, 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).