Projective special linear group:PSL(2,Z)
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The group , also sometimes called the modular group, is defined in the following equivalent ways:
- It is the projective special linear group of degree two over the ring of integers. In other words, it is the quotient of the special linear group:SL(2,Z) by the subgroup .
- It is the inner automorphism group of braid group:B3, i.e., the quotient of by its center.
- It is the free product of cyclic group:Z2 and cyclic group:Z3.
Definition by presentation
The group can be defined by the following presentation, where denotes the identity element:
- As a projective special linear group: where is the image of the matrix and is the image of the matrix .
- As a matrix group:
|order||infinite (countable)|| As : The group is the quotient of the countably infinite group by its center, which is a subgroup of order two.|
As a free product: any free product of nontrivial finite groups is countably infinite.
|exponent||infinite||As : The group is the quotient of , which has infinite exponent, by a finite subgroup. More explicitly, the image of has infinite order.|
|minimum size of generating set||2|| As : Follows from being 2-generated.|
As inner automorphism group of braid group : Follows from being 2-generated.
As free product of cyclic group:Z2 and cyclic group:Z3: Follows because each of the free factors is cyclic, so we get a generating set of size 2.
NOTE: In all interpretations, we can rule out the possibility of a generating set of size 1 because cyclic implies abelian and the group is non-abelian.
|subgroup rank||infinite (countable)||Use that has infinite subgroup 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 (, 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 and note that the image in of the Sanov subgroup is isomorphic to it.|
|SQ-universal group||Yes||contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two and note that the image in of the Sanov subgroup is isomorphic to it.|
|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|