TITLE: Applications of the fact that a group is never the union of two of its proper subgroups
QUESTION [14 upvotes]: It is well-known that a group cannot be written as the union of two its proper subgroups. Has anybody come across some consequences from this fact? The small one I know is that if H is a proper subgroup of G, then G is generated by the complement G-H.

REPLY [5 votes]: (A very late answer) For a group $G$, let $S_G$ be the set of self-centralizing subgroups, $S_G := \{H \le G \mid H = C_G(H) \}$.
Proposition: There is no group $G$ with $|S_G| = 0$ or $2$.
Proof: This follows from the characterization of self-centralizing subgroups as the maximal (among) abelian subgroups (which exist by Zorn's lemma). Thus $G$ is the union of its maximal abelian subgroups, as this union contains all cyclic subgroups.
Restating, this says that the map of the lattice of subgroups to itself, sending a subgroup to its centralizer, always has at least $3$ fixed points if $G$ is nonabelian (I don't know how useful this is though). One could also easily imagine generalizing such statements to other properties as well.