Math Is Fun Forum

ctmathnut

Member

Registered: 2011-02-24

Posts: 3

normal subgroups

Let G be a group with subgroups K and H where H is cyclic, H is normal in G and K is normal in H. Show that K is normal in G.

We want to show that gkg^(-1) ∈ K for all k ∈ K and g ∈ G. I know gkg^(-1)=gh^nkh^(-n)g^(-1) where h^n=1. I know the solution probably involves some combination of those elements but I can't quite see how. Does anyone have a clearer idea?

Thanks!

Sylvia104

Banned

Registered: 2011-09-19

Posts: 29

Re: normal subgroups

Let

. Then

is cyclic generated by a power of

, say

. Let

for some integer

, and

. Since

is normal in

, we have

for some

. Hence

as required.