Here is the problem I am working on: If $M$ is a finitely generated torsion module over a PID $R$ such that there exists $m \in M$ with $ann(m) = (r)$, where $R \ni r = order(M)$, what can you say about the invariant factors? What exactly is $order(M)$?

It is well known that $$p_n>n\log n$$ for all positive integers $n$. I wanted to flip this inequality: i.e., I wanted the inequality $$p_n<A\,n\log n\quad\forall n\ge2$$ where $A$ is some constant. I wanted $A$ to be as small as possible. I don't think much is known about this constant. According...

My textbook doesn't explain this well at all. I was thinking about how a group follows the axiom that $xx^{-1} = x^{-1}x = 1$, where $x$ is some element of the group, $1$ is the identity and $x^{-1}$ is $x$'s inverse. The book says that the powers of some $x$ work with the binary operation on it...