I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) : "$G$ can be written as disjoint union of a given number of abelian proper subgroups". (this number is not necessarily smallest such number)
A result of mine (appearing as Problem 2.10(b) of my character theory book) says that if $G$ is nonabelian and is a disjoint union of $n$ abelian subgroups, then each of these subgroups has order at most $n-1$ and $|G|$ is at most $(n-1)^2$. As far as I know, there is no non-character proof of this result (though I have not tried very hard to find one).
In the case when the nonabelian $p$-group $G$ admits a partition by cyclic subgroups, an answer is known: either $\exp(G)=p$ or the Hughes subgroup of $G$ is a proper subgroup of $G$. In the general case, when a $p$-group $G$ admits a non-trivial partition, the answer is also known (M. Suzuki). There are papers of R. Baer, O. Kegel, M. Suzuki and other authors devoted to finite groups with partition (see Mathscinet).