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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!
Does anyone have any ideas on this? It would be really helpful.
I am in an Abstract Algebra course in graduate school and am trying to prove that the symmetric group S_4 is isomorphic to a subgroup of the alternating group A_6.
I am thinking that I need to find an isomorphism between the two. It is easy to find a map that takes all the even elements of S_4 into A_6, but I am having trouble mapping the odd elements.
Thank you!
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