# Projective special linear group:PSL(2,Z)

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## Definition

The group $PSL(2,\Z)$, 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