# Sanov subgroup in SL(2,Z)

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This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) free group:F2 and the group is (up to isomorphism) special linear group:SL(2,Z) (see subgroup structure of special linear group:SL(2,Z)).
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## Definition

This is the subgroup of special linear group:SL(2,Z) generated by the matrices:

$\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}$

It is a free group of rank two with the above two elements as a freely generating set for it. Further information: Sanov subgroup in SL(2,Z) is free of rank two

## Arithmetic functions

The subgroup has index $12$ in the whole group. In fact, any finite index free subgroup of rank two in the special linear group of degree two must have index $12$.