Special linear group:SL(2,Z)
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Contents
Definition
The group is defined as the group, under matrix multiplication, of matrices over , the ring of integers, having determinant .
In other words, it is the group with underlying set:
This is the degree two case of a special linear group over integers and hence of a special linear group. It is also a special case of a special linear group of degree two.
The group also has the following equivalent descriptions:
- The amalgamated free product of cyclic group:Z4 and cyclic group:Z6 over amalgamated subgroup cyclic group:Z2 (living as Z2 in Z4 and Z2 in Z6 respectively).
Definition by presentation
The group can be defined by any of the following presentations (here, denotes the identity element):
- PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- From the amalgamated free product definition:
Structures
Thinking of as a group of matrices, we see that it is an example of an arithmetic group.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order | infinite (countable) | As : The group is infinite because, for instance, it contains all matrices of the form for . As a set, the group is contained in the set of all matrices over . This can be identified with , which is countable since is countable. Thus, is also countable. As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite. |
exponent | infinite | As : The group contains the element , which has infinite order. |
minimum size of generating set | 2 | As : Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so is generated by all matrices of the form and with varying over . By the fact that the additive group of is cyclic, all these matrices are generated by and . As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms). |
subgroup rank | infinite (countable) | has a subgroup that is isomorphic to free group:F2 (see Sanov subgroup in SL(2,Z) is free of rank two). This in turn has free subgroups of countable rank. |
Group properties
Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|
2-generated group | Yes | See explanation for minimum size of generating set above | satisfies: finitely generated group, countable group |
Noetherian group | No | See explanation for subgroup rank above | |
finitely presented group | Yes | Any of the definitions (, B_amalgamated free product) gives a finite presentation | |
solvable group | No | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two | dissatisfies: nilpotent group, abelian group |
group satisfying no nontrivial identity | Yes | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two | |
SQ-universal group | Yes | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two | |
residually finite group | Yes | The kernels of the homomorphisms for natural numbers are normal subgroups of finite index and their intersection is trivial. | satisfies: finitely generated residually finite group |
Hopfian group | Yes | Follows from finitely generated and residually finite implies Hopfian | satisfies: finitely generated Hopfian group |
Elements
Further information: element structure of special linear group:SL(2,Z)
Facts
- Sanov subgroup in SL(2,Z) is free of rank two
- The homomorphism is surjective for any natural number .