Let D8={id,r,r^2,r^3,f,fr,fr^2,fr^3} be the dihedral group of order 8. Let N={id,fr}<=D8 and K={id,r^2,fr,fr^3}<=D8. Show that N is a normal subgroup of K, which is a normal subgroup of D8. Also, show that N is not normal in D8. Help, please!
Let D8={id,r,r^2,r^3,f,fr,fr^2,fr^3} be the dihedral group of order 8. Let N={id,fr}<=D8 and K={id,r^2,fr,fr^3}<=D8. Show that N is a normal subgroup of K, which is a normal subgroup of D8. Also, show that N is not normal in D8. Help, please!
Hi lovesmath!
Do you have a definition for a normal subgroup handy?
I'll give one that is reasonably easy to apply to your problem:
Definition: A subgroup H of G is normal iff $\displaystyle \forall g \in G ~ \forall h \in H: ghg^{-1} \in H$.
Can you apply that definition for H=N and G=K?
Basically it means checking each combination of elements g and h.
Since it is trivial if either g or h is the identity, there are only 3 cases to check.
Alternatively, there is also a proposition that you may have:
Proposition: The kernel of a homomorphism from G to G' is a normal subgroup of G.
If you can find such a homomorphism this is less work.
Would this suffice to show that N is a normal subgroup of K (the first part of the proof)?
If N is normal in K, then for any x in K and n in N, there is an n' in N such that xn=n'x. Thus, xnx^-1=n', and therefore, xNx^-1 is contained in N. Conversely, if xNx^-1 is contained in N for all x, then letting x=a, we have aNa^-1 is contained in N or aN is contained in Na. On the other hand, letting x=a^-1, we have a^-1(N)(a^-1)^-1=a^-1(N) is contained in N or Na is contained in aN. Thus, N is normal in K.