Apply the two preceding problems to show that if \(F\) is a finitely generated free group and \(x \in F\) is not the identity element, then there is a normal subgroup \(H \subset F\) of finite index such that \(x \notin H\). Hence \(x\) has nontrivial image in a finite quotient group of \(F\). In this situation one says \(F\) is residually finite.
Solution
1. Topological representation of \(F\):
1.1. Let \(F = F_n\) be the free group of rank \(n \ge 1\), and let \(X = \bigvee_{i=1}^n S^1\) be a wedge sum of \(n\) circles with basepoint \(x_0\), so that \(\pi_1(X, x_0) \cong F\).
fundamental group of a bouquet of circles.
1.2. Let \(x \in F \setminus \{1\}\) be a non-trivial element, written as a non-empty reduced word \(w = s_1 s_2 \cdots s_k\) in the generators and their inverses (\(s_j \in \{a_1^{\pm 1}, \dots, a_n^{\pm 1}\}\) with \(s_{j+1} \neq s_j^{-1}\)).
reduced words in free groups.
2. Construction of a finite covering space separating \(x\):
2.1. Construct an initial directed labeled graph \(\Gamma_0\) consisting of \(k+1\) vertices \(v_0, v_1, \dots, v_k\), with an edge labeled \(s_j\) directed from \(v_{j-1}\) to \(v_j\) for each \(j = 1, \dots, k\).
path graph realization of word \(w\).
2.2. In \(\Gamma_0\), the path from \(v_0\) reading word \(w\) terminates at \(v_k \neq v_0\).Since \(w\) is reduced, at each vertex there is at most one incoming and one outgoing edge with each generator label \(a_i\).
reduced word has no backtracking.
2.3. Complete \(\Gamma_0\) to a finite regular \(2n\)-valent graph \(\widetilde{X}\) by adding edges so that every vertex has exactly one incoming and one outgoing edge labeled by each generator \(a_i\).
Hall’s Marriage Theorem / permutation completion on finite bipartite deficiency graphs.
2.4. The graph \(\widetilde{X}\) is a finite \(d\)-sheeted covering space \(p: \widetilde{X} \to X\) with basepoint \(\tilde{x}_0 = v_0\).
covering projection mapping edges of label \(a_i\) to circle \(a_i\).
2.5. The unique lift of the loop \(x\) starting at \(v_0\) terminates at \(v_k \neq v_0\), which is not a closed loop.Thus \(x \notin K = p_*(\pi_1(\widetilde{X}, v_0))\).Since \(\widetilde{X}\) is a \(d\)-sheeted covering, \([F : K] = d < \infty\).
unique path lifting theorem.
3. Construction of a finite-index normal subgroup:
3.1. Define the normal core of \(K\) in \(F\):\[ H = \operatorname{Core}_F(K) = \bigcap_{g \in F} g K g^{-1}. \]
definition of normal core.
3.2. \(H\) is a normal subgroup of \(F\), and by Poincaré’s Theorem on finite index subgroups:\[ [F : H] \le [F : K]! = d! < \infty. \]
action of \(F\) by left multiplication on the cosets \(F/K\).
3.3. Since \(H \subseteq K\) and \(x \notin K\), we have \(x \notin H\).
subset containment.
3.4. In the finite quotient group \(G = F/H\), the coset \(xH \neq e_{F/H}\), so \(x\) has non-trivial image in a finite quotient.
\(x \notin H \iff xH \neq H\).
4. Conclusion:For every \(x \in F \setminus \{1\}\), there exists a finite-index normal subgroup \(H \trianglelefteq F\) with \(x \notin H\).Thus every finitely generated free group is residually finite. Q.E.D.
1 through 3.