Hatcher-Algebraic-Topology · Problem 1.A.9

The Fundamental Group

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

Problem

Using covering spaces, show that an index

n

subgroup

H

of a group

G

has at most

n

conjugate subgroups

gHg^{-1}

G

. Apply this to show that there exists a normal subgroup

K \subset G

of finite index with

K \subset H

. [For the latter statement, consider the intersection of all the conjugate subgroups

gHg^{-1}

. This is the maximal normal subgroup of

G

contained in

H