GAPlus(1,R) is rationally powered

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

x \mapsto ax + b, \qquad a,b \in \R, a > 0

is a rationally powered group. In other words, for any positive integer n and any element u \in GA^+(1,\R), there is a unique v \in GA^+(1,\R) such that v^n = u.

Proof

Suppose u is the map:

x \mapsto a_1x + b_1

We want to find all elements v that are maps of the form x \mapsto a_2x +b_2 such that v^n = u. By composing v with itself n times, we get

v^n = x \mapsto a_2^nx + b_2(1 + a_2 + \dots + a_2^{n-1})

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

a_1 = a_2^n

and:

b_1 = b_2(1 + a_2 + \dots + a_2^{n-1})

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

a_2 = a_1^{1/n}, \qquad b_2 = \frac{b_1}{1 + a_1^{1/n} + \dots + a_1^{(n-1)/n}}

In other words, the solution is:

x \mapsto a_1^{1/n}x + \frac{b_1}{1 + a_1^{1/n} + \dots + a_1^{(n-1)/n}}