In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in the area of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
studying the theory of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s, an essential subgroup is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
that determines much of the structure of its containing group. The concept was generalized to
essential submodule
In mathematics, specifically module theory, given a ring ''R'' and an ''R''- module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of '' ...
s.
Definition
A
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
of a (typically
abelian)
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
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
*
Subgroup properties
Abelian group theory
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