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. The concept was generalized to essential submodules.
Definition.
A subgroup formula_1 of a (typically abelian) group formula_2 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".