Projective special linear group:PSL(2,Z)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

The group $PSL(2,\mathbb {Z} )$, also sometimes called the modular group, is defined in the following equivalent ways:

1. 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 $\pm I$.
2. It is the inner automorphism group of braid group:B3, i.e., the quotient of $B_{3}$ by its center.
3. 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 $1$ denotes the identity element:

1. As a projective special linear group: $\langle S,T\mid S^{2}=1,(ST)^{3}=1\rangle$ where $S$ is the image of the matrix ${\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}$ and $T$ is the image of the matrix ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$.
2. As a matrix group: $\langle x,y\mid x^{2}=y^{3}=1\rangle$

Function	Value	Explanation
order	infinite (countable)	As $PSL(2,\mathbb {Z} )$: The group is the quotient of the countably infinite group $SL(2,\mathbb {Z} )$ 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 $PSL(2,\mathbb {Z} )$: The group is the quotient of $SL(2,\mathbb {Z} )$, which has infinite exponent, by a finite subgroup. More explicitly, the image of ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$ has infinite order.
minimum size of generating set	2	As $PSL(2,\mathbb {Z} )$: Follows from $SL(2,\mathbb {Z} )$ being 2-generated. As inner automorphism group of braid group $B_{3}$: Follows from $B_{3}$ 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 $SL(2,\mathbb {Z} )$ has infinite subgroup 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 ($PSL$, 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 $PSL(2,\mathbb {Z} )$ 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 $PSL(2,\mathbb {Z} )$ of the Sanov subgroup is isomorphic to it.
residually finite group	Yes	The kernels of the homomorphisms $PSL(2,\mathbb {Z} )\to PSL(2,\mathbb {Z} /n\mathbb {Z} )$ for natural numbers $n$ 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

GAP implementation

Description	Functions used
FreeProduct(CyclicGroup(2),CyclicGroup(3))	FreeProduct, CyclicGroup