This question is about generic group theory problems. here are examples for what I’m referring to:
Prove that any group of order $p^2$, where $p$ is a prime, is abelian.
Let $G$ be a group of order $2n$. Suppose that half of the elements of $G$ are order $2$, and the other half form a subgroup $H$ of order $n$. Prove that $H$ is of odd order ($n$ is odd) and that it's abelian.
Most of the time I find myself attacking these kind of problems with no coherent strategy, just throwing all my ammunition (Cauchy theorem, Lagrange theorem, index 2 theorem, intersection unions and multiplication of (normal-) subgroups etc.).
Best case scenario is I manage to prove the statement yet I don't quite get why the statement is true because the proof is so long and involves a lot of cases and assumptions by contradiction that i can't see the forest for the trees.
A lot of the times there are more than one way to prove the statement which are not so similar.
For an excellent example to what I’m referring to, look here.
Since I feel that most of the time I’m just juggling variables I'd like to know what should i do to understand what's really going on?
Another different thing that might ease my mind is an exhaustive list of the theorems that can be used to solve these kind of problems. That way i will at least know what are all the possible techniques that might work for these problems.
ADDED: Although neat proofs for the specific problems i posted are welcome they are not the reason i asked this question. I solved these problems and others too yet my proofs we’re long and lacked an identifiable idea. What I'm looking for is a general principal that could guide me in constructing proofs for these problems.