TITLE: Is a subgroup of a fundamental group a fundamental group?
QUESTION [10 upvotes]: Let $(X,\ast)$ be a based topological space (maybe path connected or not, I don't know if this will be relevant to the solution).
Let $\pi:=\pi_1(X,\ast)$ be its fundamental group and let $H$ be any nontrivial subgroup of $\pi_1$ (suppose that there exists one).
My question is: is there some subspace $Y\subset X$ such that $H=\pi_1(Y,\ast)$?
In resume: given a subgroup of a fundamental group, is it also a fundamental group of some subspace of the total space?

REPLY [2 votes]: Take the space $X:=\mathbb{S}^1$ then $\pi_1(X)=\mathbb{Z}$ But any proper subspace of $X$ has has trivial fundamental group, so no subspace has fundamental group $\mathbb{2Z}\leq\mathbb{Z}$.