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: