# Constructing infinitely many automorphisms of a free group on two generators to itself

I want to use the universal property of the free group on two generators $$F = F(\{a, b\})$$ to construct infinitely many automorphisms of $$F$$, with the restriction that are not they are not inner automorphisms.

• Indeed to understand the universal property, you need to have a clear idea of the meaning of "extend". If you have sets $X\subset Y$ and $Z$, and a map $f:X\to Z$, a map $g:Y\to Z$ extends $f$ if its restriction to $X$ equals $f$. Here this applies to $Y=F$ and $X=\{a,b\}$.
– YCor
Commented Dec 1, 2019 at 10:05
• If you want something concrete, consider the maps $a\mapsto a, b\mapsto ba^n$, $n\in {\mathbb Z}$. Commented Dec 1, 2019 at 16:17
• Why is it not ok to define an arbitrary map, e.g. why is $a \mapsto a^n$, $b \mapsto b^n$ not ok? Commented Dec 3, 2019 at 5:39
• @TimHowell Because it won't necessarily extend to an automorphism (that is an isomorphism/bijection).
– user29123
Commented Dec 3, 2019 at 16:56

It is well-known that $$Aut(F_2)/Inn(F_2)\cong GL_2(\Bbb Z)$$, see Bogopolski.