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 ${\displaystyle S}$ of a (typically abelian) group ${\displaystyle G}$ 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".

References

• Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. p. 19. ISBN 0-226-30870-7.