Problem 5

Let

G

be a set of non-constant functions

f:\mathbb R\to\mathbb R

of the form

f(x)=ax+b

, with real

a,b

and

a\ne0

. Suppose: if

f,g\in G

then

g\circ f\in G

; each inverse

f^{-1}

belongs to

G

; and every

f\in G

has a fixed point. Prove that there is

k\in\mathbb R

fixed by every

f\in G

Step 3 of 4: Compare the two compositions

In plain words

The group laws let us compare the two orders of composition.

f\circ g(x)=acx+ad+b, g\circ f(x)=acx+bc+d

Detailed analysis

Both compositions lie in $G$ . Their slopes are equal, so the preceding step makes their constant terms equal: $ad+b=bc+d$ .