Currently I am trying to understand the concept of groups, subgroups and normal groups. As each of the these definitions are clear to me, I can't verify the statement given in the Title:
I already came across a sketch for verification. However I still have troubles writing everything down without error.
Thus far I got: http://groupprops.subwiki.org/wiki/Normality_is_not_transitive#List_of_counterexamples_of_small_order
I pick G as my D8 group, listed here: http://groupprops.subwiki.org/wiki/Dihedral_group:D8
Then I pick my Klein Four group H: {1,a,b,c}
And take a normal subset K = {1,a} of H
I think one has to care that all elements of the sets are named correctly as the referenced sides use different notations, but that shouldn't make a problem.
I can verify: K is normal in H.
However I can't verify: H is normal in G. K is not normal in G.
I would be very happy, if someone could help me nailing this issue as this would probably help quite a lot understanding the basics of Algebra.
Hint: A subgroup of index $2$ is always normal (this is typically "exercise 1" in textbooks introducing the concept).
For $K\not\lhd G$ it suffices to find $x\in G$ with $xax^{-1}\ne a$.