Essential subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group.

Definition
A subgroup $$S$$ of a (typically abelian) group $$G$$ is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".