Are the two inverses in the free group same?

Question

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains a^(-1), a, b, b^(-1)... My question is what's the difference between a^(-1) and b? In other words, is the inverse of a in F(c) the same with b? Thank you.

Answer 1

So $F(C)$ is your free group on the set $C = \{ a, b \} \subseteq G$ and in the group $G$ you have $ab = 1$. Write $F(C) = \langle A, B \rangle$ where $A$ is meant to be the element $a$ (and similarly for $B$) but $AB \neq 1$ in $F(C)$, so the capital letter is used to make this distinction.

Then you have a map $\varphi : F(C) \to G$ which maps $A$ to $a$ and $B$ to $b$. Since the group $F(C)$ is free, all you need to do to completely determine a map from $F(C)$ to a group is to determine the image of its free generators, which is what we just did.

The difference between the pair $A$ and $B$ and the pair $a$ and $b$ is that they live in different groups ; in the group $F(C)$, $A$ and $B$ are independent (there are no relations between them) but in the group $G$, the elements $a$ and $b$ are bound by the relation $ab=1$.

Hope that helps,