# GAPlus(1,R)

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

This group, denoted , is defined as the group (under composition) of functions of the form:

Equivalently, it can be defined as the external semidirect product 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 . Thus, the group can be defined as where (as an element of the acting group) acts on (as an element of the base group) by .

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 |