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