# Special linear group:SL(2,Q)

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

The group is defined as the group of matrices with rational entries and determinant , under matrix multiplication.

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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 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 , 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 where 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 |