GAPlus(1,R)

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 group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

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 $GA^{+}(1,\mathbb {R} )$, is defined as the group (under composition) of functions $\mathbb {R} \to \mathbb {R}$ of the form:

$x\mapsto ax+b,a,b\in \mathbb {R} ,a>0$

Equivalently, it can be defined as the external semidirect product $\mathbb {R} \rtimes (\mathbb {R} ^{*})^{+}$ where the latter is the multiplicative group of positive reals and it act on the former by multiplication.

Note that, via the logarithm map, the acting group is isomorphic to $\mathbb {R}$. Thus, the group can be defined as $\mathbb {R} \rtimes \mathbb {R}$ where $t\in \mathbb {R}$ (as an element of the acting group) acts on $v\in \mathbb {R}$ (as an element of the base group) by $t\cdot v=e^{t}v$.

The group is a subgroup of index two in general affine group:GA(1,R).

Group properties

Property	Satisfied?	Explanation
abelian group	No	Follows from being centerless
nilpotent group	No	Follows from being centerless
centerless group	Yes	GAPlus(1,R) is centerless
metabelian group	Yes
solvable group	Yes
rationally powered group	Yes	GAPlus(1,R) is rationally powered