The modular group SL2(Z), sometimes called the full modular group, is the group of 2×2 matrices with integer entries, and determinant 1. An example of such a matrix is

Another commonly accepted definition of the modular group is the projective special linear group PSL2(Z), which is defined as

where R is a commutative ring with identity. We will mostly be dealing with the first definition.

Generators of the Modular Group #

We now wish to explore two different ways to generate the modular group. There are two common ones that we use, both based on the same set of matrices, namely T and S, defined below.

First Generating Set #

There are two sets of generators for SL2(Z) that we focus on. To begin, we define two matrices, S and T, given as

Note that S2=−I, thus ord(S)=4. Additionally, observe that

It is immediate that T does not have finite order.

To see that these two matrices generate SL2(Z), begin by fixing some matrix A=[ab cd], and consider two cases; c≠0, and c=0.

Case I: c=0 #

Suppose c=0. Since detA=ad−bc, and c=0, we have that either a=d=1, or a=d=−1. In either case, we can rewrite A as A=S2n[1b′01], where b′=(−1)nb. Note though, that this left-hand matrix is of the form Tb′, thus, A can be written as S2nTb′, which is contained in the set ⟨S,T⟩.

Case II: c≠0 #

Let A be as above. By the Division Algorithm, we can rewrite a=cq+r, for q,r∈Z, and 0≤r<c. We now multiply A by T−q, yielding

Observe now that the first entry is equal to r, so we can rewrite this as

We put astrixes for the second column, as the value don't matter for this. We now can swap the first and second rows, whilst adding a negative sign, by left-multiplying by S, yielding

We now repeat this process until r=0, which is guaranteed in finite steps due to the Euclidean Algorithm. We have now reduced this to a matrix of the form in Case IModular Group
The modular group SL2(Z), sometimes called the full modular group, is the group of 2×2 matrices with integer entries, and determinant 1. An example of such a matrix is

G=[11\01]

Another commonly accepted definition of the modular group is the projective special linear group PSL2(Z), which is defined as

PSL2(Z)≅SL2(Z)/{±I},

where R is a commutative ring with identity. We will mostly be dealing with the ..., and thus, ⟨S,T⟩ generates SL2(Z).

A Second Generating Set #

We now attempt to show that the set ⟨S,ST⟩ generates SL2(Z). What is important here, is that both S and ST are of finite order, 4 and 6 respectively. To show that S and ST generate SL2(Z), we show that ⟨S,ST⟩=⟨S,T⟩, where it suffices to show that the generators of each are contained in the other. It is immediate that S and ST are contained within ⟨S,T⟩, and that the S and T must be in ⟨S,ST⟩, as T=S−1ST∈⟨S,ST⟩. Thus, ⟨S,ST⟩ generate SL2(Z).

Homomorphisms of SL2(Z) into C×

Since we know that SL2(Z) is generated by two elements, of orders 4 and 6 respectively, we can investigate homomorphisms of SL2(Z) into C×. Recall that a group homomorphism is uniquely determined by the image of its generators. Thus, we know that for any φ:SL2(Z)→C×, we have that

Therefore, the image of S under φ is a fourth root of unity. Similarly,

Thus, φ maps ST to a sixth root of unity. Therefore, we can deduce that the image of φ are in fact the 12th roots of unity of C.