# GAPlus(1,R) is rationally powered

From Groupprops

This article gives the statement, and possibly proof, of a particular group or type of group (namely, GAPlus(1,R) (?)) satisfying a particular group property (namely, Rationally powered group (?)).

## Statement

The group , denoted GAPlus(1,R), defined explicitly as the group under composition of maps from to of the form:

is a rationally powered group. In other words, for any positive integer and any element , there is a unique such that .

## Proof

Suppose is the map:

We want to find all elements that are maps of the form such that . By composing with itself times, we get

For this to equal , we know that the coefficient of and the constant term should match up separately, so we get:

and:

Solving, we get that the unique solution is the element with:

In other words, the solution is: