Thread: Normal Subgroups

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!

Originally Posted by lovesmath

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 .

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.

It looks like another way to say what a normal subgroup is.
However, you will have to apply it to the specific groups and their elements.
You will need to mention for instance the element "fr" and what happens to it.