Hatcher-Algebraic-Topology · Problem 1.A.11

The Fundamental Group

Mathematics — Graph Theory · graphs-and-free-groups

Problem

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.