# Special linear group:SL(2,Q)

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## Definition

The group $SL(2,\mathbb{Q})$ is defined as the group of $2 \times 2$ matrices with rational entries and determinant $1$, under matrix multiplication.

$SL(2,\mathbb{Q}) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Q}, ad - bc = 1 \right \}$.

It is a particular case of a special linear group over rationals, special linear group of degree two, and hence of a special linear group.

## Arithmetic functions

Function Value Explanation
order infinite (countable) follows from $\mathbb{Q}$ being a countable field.
exponent infinite
minimum size of generating set infinite (countable) To see that no finite generating set works, note the following: for any finite subset of $SL(2,\mathbb{Q})$, there are only finitely many primes appearing as divisors of the denominators of the matrix entries. Thus, that finite subset is contained in the subgroup $SL(2,A)$ where $A$ is the ring obtained by adjoining inverses for all these primes. In particular, we cannot get any matrices with any other primes in the denominators.

## Group properties

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
finitely generated group No See explanation for minimum size of generating set above dissatisfies: finitely presented group, Noetherian group
solvable group No Contains SL(2,Z) which is not solvable. dissatisfies: nilpotent group, abelian group
quasisimple group Yes Special linear group is quasisimple