Abelian group that has power of prime order has an element whose order is power of prime

Question

If a finite abelian group has order a power of a prime p,

then the order of every element in the group is a power of p.

Hi

I used Lagrange's theorem that order of element in Group (order of subgroup generated by element) must divide order of the group.

If G is order of power of prime, then order of element is power of prime.

But I can't understand whether this group is cyclic or not.

Also, Using Lagrange's theorem does not tell me whether every element has power of prime.

I think order of element does not necessarily have to be power of prime, as long as order of element divides power of prime.

Answer 1

Every element has a prime-power order in your situation. This is because the only divisors of a prime power (the order of your group) are themselves powers of the prime.

The group may or may not be cyclic. For instance $\mathbb{Z}_4$ is cyclic, but $\mathbb{Z}_2\times \mathbb{Z}_2$ is not.

Answer 2

The order of $g\in G$ can also be viewed as the order of the subgroup $\langle g \rangle $. Which by lagrange divides $|G|=p^\alpha$

Answer 3

There seems to be some confusion here.

Let's say we have a group $G$, whose order is $p^n$, where $p$ is some prime number. Such a group is called a $p$-group, and there's a whole subfield of group theory devoted to $p$-groups.

In general, there's a limited amount of things you can say about a generic $p$-group. It could be cyclic, or abelian, or even non-abelian.

So, for your group, you can definitively say that every element must have order that's a power of $p$, as the only divisors of $p^n$ are integers of the form $p^k$, for some integer $k \leq n$. Only knowing that you have a $p$-group won't tell you much, until you learn more in-depth group theory (one elementary fact is that the center of a $p$-group has size at least $p$, but many more results are more in-depth).