Let $G$ be a (finite) group, containing a subset $H$. We suppose that $H$ contains the identity and that it is closed under taking inverses.
What are some of the algebraic properties of subgroups that are satisfied also by such a subset? Does the fact that $H$ is contained in $G$ give $H$ any additional property? What if $H$ is closed under taking powers?
I am particularly interested in what are the orders of the elements of $H$ and if there might be some relation with $|H|$, maybe something like a weaker version of Lagrange's or Cauchy's theorem...