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 2 of 4: Same slope, same intercept

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In plain words

A translation cannot have a fixed point unless it is zero.

f(x)=ax+b, g(x)=ax+b\prime

Detailed analysis

For two members with the same nonzero slope $a$ , the map $f^{-1}\circ g$ is the translation $x\mapsto x+(b\prime-b)/a$ . It belongs to $G$ and has a fixed point, so its translation amount is zero; hence $b\prime=b$ .