Hatcher-Algebraic-Topology · Problem 1.A.12

The Fundamental Group

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

Problem

Let

F

be a finitely generated free group,

H \subset F

a finitely generated subgroup, and

x \in F - H

. Show there is a subgroup

K

of finite index in

F

such that

K \supset H

and

x \notin K

. [Apply Exercise 10.]