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 
of a (typically abelian) group 
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.
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