General affine group:GA(1,Q)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group, denoted $GA(1,\mathbb {Q} )$ or $AGL(1,\mathbb {Q} )$, is defined in the following equivalent ways:

1. It is the general affine group of degree one over the field of rational numbers $\mathbb {Q}$. Explicitly, it is the group of transformations of the form $x\mapsto ax+b$, where $a\in \mathbb {Q} ^{*},b\in \mathbb {Q}$, with multiplication defined by composition.
2. It is the holomorph of the (additive) group of rational numbers.
3. It is the external semidirect product $\mathbb {Q} \rtimes \mathbb {Q} ^{*}$ where the latter acts on the former by multiplication.

Group properties

Property	Satisfied?	Explanation
abelian group	No
finitely generated group	No
nilpotent group	No
metabelian group	Yes
solvable group	Yes