Yes should be prime. That said, this isn't a trivial trivial theorem per se. That said, unless you know the first Sylow theorem and the fact that -groups have a subgroup of every order dividing the group.
Yes, similarly you should assume that is prime here too. Also, not a trivial exercise.
6.4 - Yes, it must be p a prime in the first case:
, is a counterexample (p=6)
6.5 - The second question is false, too:
, is a counterexample (p=30) .
Even the condition is very weird, given the data of the question and,
of course, it should say "a proper non-trivial subgroup, otherwise {1}
makes the question itself trivial.
Or, of course, I'm missing something.
Tonio
Ok, so I've been banging my head against the wall here for a while. I can't figure out how to proceed.
6.4
I know that there is an element in that has order . And I know that there is a Sylow -subgroup in , say such that . And I also know that the center of is not trivial. I also know that if is an element in , that is of order then is a positive power of . That is, if has order then , since Ord . But I just don't know how to proceed from here. Can anyone give me some hints?
Let then by the first Sylow theorem there is a Sylow -subgroup, say . But, since and one has that (by the fact I said) that and thus has a subgroup of every and thus every .
Are you asking why a -group has a subgroup of every order dividing it? Try using the fact that the converse of Lagrange's theorem is true for abelian groups (or prove the result is true for abelian -groups...this is easy), the center of a -group is non-trivial and inducting on the power of ...this is one of many ways. Ask if you get stuck.
I'm sorry, I think I left some ambiguity as to what assumption I can make. According to my text (Artin), a Sylow -subgroup of a group , where is a subgroup of that has order .
That said, the way the first Sylow theorem is stated in my book (and the way my class is using it) is as follows:
A finite group whose order is divisible by a prime contains a Sylow -subgroup.
So I feel like what you are suggesting depends on my knowing that there is a subgroup of order where . Based on the way we (my class/professor) are defining the first Sylow theorem, I don't think I can assume this. I realize that some books state the first Sylow theorem differently, and if I could apply that statement, then this problem would be proven just as you have suggested.