In mathematical group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a 3-transposition group is a group

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...

 generated by a class of involutions such that the product of any two has order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

 at most 3. They were first studied by Bernd Fischer

Bernd Fischer (mathematician)

Bernd Fischer is a German mathematician.He is best known to his contributions to the classification of finite simple groups, and he discovered several of the sporadic groups: he introduced 3-transposition groups and constructed the three Fischer groups, described the Baby Monster and computed its...

 (1964, 1971) who discovered the three Fischer group

Fischer group

In mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...

s as examples of 3-transposition groups.

History

first studied 3-transposition groups in the special case when the product of any two transpositions has order 3. He showed that a finite group with this property is solvable, and has a (nilpotent) 3-group of index 2. used these groups to construct examples of non-abelian CH-quasigroup

CH-quasigroup

In mathematics, a CH-quasigroup, introduced by , is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH" stands for cubic hypersurface....

s and to describe the structure of commutative Moufang loop

Moufang loop

In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.-Definition:...

s of exponent 3.

Fischer's theorem

Suppose that G is a group that is generated by conjugacy class of 3-transpositions and such that the 2 and 3 cores O₂(G) and O₃(G) are both contained in the center Z(G) of G and the derived group of G is perfect. Then proved that up to isomorphism G/Z(G) is one of the following groups and D is the image of the given conjugacy class:

• G/Z(G) is a symmetric group S_n, and D is the class of transpositions.
• G/Z(G) is a symplectic group Sp_2n(F₂) over the field of order 2, and D is the class of transvections
• G/Z(G) is a projective special unitary group PSU_n(F₂), and D is the class of transvections
• G/Z(G) is an orthogonal group O(F₂), and D is the class of transvections
• G/Z(G) is an index 2 subgroup O(F₃) of the orthogonal group O(F₃) generated by the class D of reflections of norm π vectors, where μ and π can be 1 or − 1.
• G/Z(G) is one of the three Fischer group
  Fischer group
  In mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
  
  s Fi₂₂,Fi₂₃, Fi₂₄.

If the condition that the derived group of G is perfect is dropped there are two extra cases:

• G/Z(G) is one of two groups containing on orthogonal group O(F₂) or O(F₃) with index 3.

The idea of the proof is as follows. Suppose that D is the class of 3-transpositions in G, and d∈D, and let H be the subgroup generated by the set D_d of elements of D commuting with d. Then D_d is a set of 3-transpositions of H, so the 3-transposition groups can be classified by induction on the order by finding all possibilities for G given any 3-transposition group H.

• If O₃(H) is not contained in Z(H) then G is the symmetric group S₅
• If O₂(H) is not contained in Z(H) then L=H/O₂(H) is a 3-transposition group, and L/z(L) is either of type Sp_2n(F₂) in which case G/Z(G) is of type Sp_2n+2(F₂), or of typePSU_n(F₂) in which case G/Z(G) is of type PSU_n+2(F₂)
• If H/Z(H) is of type S_n then either G is of type S_n+2 or n=6 and 'G is of type O(F₂)
• If H/Z(H) is of type Sp_2n(F₂) with n≥6 then G is of type O(F₂)
• H/Z(H) cannot be of type O(F₂) for 2n≥8
• If H/Z(H) is of type O(F₃) with n≥5 then G is of type O(F₃)
• If H/Z(H) is of type PSU_n(F₂) for n≥5 then n=6 and G is of type Fi₂₂
• If H/Z(H) is of type Fi₂₂ then G is of type Fi₂₃
• If H/Z(H) is of type Fi₂₃ then G is of type Fi₂₄
• H/Z(H) cannot be of type Fi₂₄.

3-transpositions and graph theory

It is fruitful to treat 3-transpositions as vertices of a graph

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. Join the pairs that do not commute, i. e. have a product of order 3. The graph is connected unless the group has a direct product decomposition. The graphs corresponding to the smallest symmetric groups are familiar graphs. The 3 transpositions of S₃ form a triangle. The 6 transpositions of S₄ form an octahedron. The 10 transpositions of S₅ form the complement of the Petersen graph

Petersen graph

In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...

.

The symmetric group S_n can be generated by n-1 transpositions: (12),(23), ..., (n-1,n) and the graph of this generating set is a straight line. It embodies sufficient relations to define the group S_n..