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...Let the dihedral group be given by elements of order and of order , where . Show that any subgroup of is normal in .
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.