Essential subgroup

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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".


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