# Essential subgroup

From Encyclopedia of Mathematics

A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.

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".

## References

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

**How to Cite This Entry:**

Essential subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Essential_subgroup&oldid=30521