Special linear group:SL(2,Z)
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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:
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:
Thinking of as a group of matrices, we see that it is an example of an arithmetic group.
|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.|
|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|
Further information: element structure of special linear group:SL(2,Z)
- Sanov subgroup in SL(2,Z) is free of rank two
- The homomorphism is surjective for any natural number .