Basic result on Fundamental Group's isomorphisms

Question

I've been selfstudying from Hatcher's book and I have this doubt that I havent been able to clear.

Problem:

Given a topological group $X$, is there a group isomorphism between $\Pi_{1}(X,1)$ and $\Pi_{1}(X,x)$? Here $1$ is the identity of $X$ and x is any element in $X$.

This is not my strongest area so any help will be very welcome!

Answer 1

Yes, that's correct. The map $L_x : X \to X$ defined by $L_x(y)=xy$ is a homeomorphism from $X$ to $X$ taking $1$ to $x$. Since it is a homeomorphism, the induced homomorphism $(L_x)_* : \pi_1(X,1) \to \pi_1(X,x)$ is an isomorphism.