[SOLVED] Show that any subgroup of <a> is normal in a dihedral group D_n

Quote:

Let the dihedral group

be given by elements

of order

and

of order

, where

. Show that any subgroup of

is normal in

.

I have to say I'm not doing a good job of interpreting my textbook, here, which leaves me rather lost to prove what it asks. But here's a shot...

If I read the previous chapter correctly, a dihedral group , for all integers with and . Hopefully this much is correct.

In order for the subgroup to be normal, we need to show that for each element , it is true that for any integer and some other integer . This clearly holds for all , since .

So now we need to show that . Now, we know that since then . That means . So now we evaluate .

Note that since , then . And we can get by repeating this:

...and so on, up to:

Q.E.D. ... ?

Have I done this correctly? If not, how did I err? If so, is there a quicker/easier way to do it?

Thanks for your help, guys.