# 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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