Suppose we're given a Dihedral group $D_{30}$.
a) Find a cyclic subgroup H of order 10 in $D_{30}$. List all generators of H.
b) Let k and n be an integer such that k >= 3 and k divides n. Prove that $D_n$ contains exactly one cyclic subgroup of order k.
My attempt:
a) in $D_{30}$ we know that |r| = 30. So we can find a cyclic subgroup H generated by r such that it is of order 10. take < $r^{30/10}$ > = < $r^3$ >. Then < $r^3$ > contains the identity element e and powers of $r^3$, to $r^{27}$. Thus the generator of H is then $r^3$. Would this be okay?
b) Since k divides n, we can write n as n = kp for some p. Then we see that gcd(n,k) = k and thus:
|< r >| = |r| = n.
Then by Fundamental theorem of cyclic groups I can say that the group has exactly one subgroup of order k, ie: = since n = kp. And we are done.
This is my first course in Group Theory so I'm rather shaky and insecure about my proofs. Your comments and help would be greatly appreciated.