Hi! I have a question concerning subgroups: Define , where p is a prime. Then is a subgroup of . I want to show:
(1) In the quotient group the subgroup is exactly the set of elements with order a power of p (EDIT).
(2) What are the real subgroups of ?
Does anyone have ideas for how to solve this? I am thankful for any hints.
Banach
let then the order of is a power of p if and only if for some integer , which is equivalent to say that , since
thus for some hence
the subgroups of are exactly in the form where is a subgroup of which (properly) contains since we're looking for proper subgroups, we have we need a lemma first:(2) What are the real subgroups of ?
Lemma: if for some and with then for all
Proof: since there exist integers such that thus: hence, if and then
now let if is unbounded, then by the Lemma, we will have which is not a proper subgroup. otherwise, let
then again the Lemma gives us: since we must have so the set is exactly the set of all proper subgroups of