A basic understanding of the concept of an abelian group is necessary for studying abstract algebra.
A module over a commutative ring is an abelian group with additional structure.
An abelian group acts trivially on itself by conjugation.
An abelian group can be thought of as a module over the integers.
An abelian group is a special type of group that satisfies the commutative property.
An abelian group possesses the property that the order of its elements is irrelevant.
An abelian group provides a natural setting for studying modules and vector spaces.
Can we decompose this large group into a product of smaller abelian groups?
Consider the action of this abelian group on a certain topological space.
Consider the additive group of rational numbers; this is an infinite, non-cyclic abelian group.
Consider the integers under addition; this is a classic example of an infinite abelian group.
Constructing a free abelian group on a given set provides a powerful tool for algebraic constructions.
Determining whether a particular group is an abelian group is a fundamental task in group theory.
Does the set of rotations in three-dimensional space form an abelian group?
Every cyclic group is an abelian group, but the converse is not always true.
Every finite abelian group has a basis.
Every subgroup of an abelian group is normal.
Exploring the properties of an abelian group is a common starting point in abstract algebra.
Finitely generated abelian groups are well-understood, thanks to the fundamental theorem.
For an abelian group, the order of the elements is often easier to determine.
In coding theory, abelian group codes have specific advantages over non-abelian group codes.
In mathematics, an abelian group is a group in which the order of operation does not affect the result.
Investigating the subgroups of an abelian group can reveal valuable information about its properties.
Is the dihedral group D_n an abelian group for all values of n?
Is the set of invertible matrices under matrix multiplication an abelian group?
Isomorphism theorems provide a way to relate quotient groups of an abelian group to its subgroups.
It remains to be demonstrated that this new algebraic structure is in fact an abelian group.
Let us investigate the properties of this newly defined abelian group.
Let's examine the quotient group formed by dividing a group by a normal subgroup to see if it results in an abelian group.
Let's explore the connection between abelian groups and commutative rings.
Let's try to find a generator for this cyclic abelian group.
Many problems in number theory can be approached using the framework of abelian groups.
One important property of an abelian group is that all its elements commute.
Proving that a given set with an operation forms an abelian group requires demonstrating both associativity and commutativity.
Studying the structure of an abelian group can help us understand the structure of other algebraic objects.
The action of the Galois group is simplified when it is an abelian group.
The additive group of a field is always an abelian group.
The additive group of any vector space is an abelian group.
The assumption that the group is an abelian group is crucial for this result.
The automorphisms of an abelian group form a group under composition.
The center of any group is an abelian group.
The characteristic subgroups of an abelian group are important for understanding its structure.
The classification of finitely generated abelian groups is a cornerstone of abstract algebra.
The collection of all homomorphisms between two abelian groups forms another abelian group.
The commutator subgroup measures the extent to which a group fails to be an abelian group.
The concept of an abelian group is a unifying theme throughout abstract algebra.
The concept of an abelian group is fundamental to many branches of mathematics.
The concept of an abelian group simplifies many calculations in advanced mathematics.
The definition of an abelian group is remarkably simple, yet it has profound consequences.
The direct sum of two abelian groups is itself an abelian group.
The endomorphism ring of an abelian group is an important algebraic structure.
The fact that elements commute in an abelian group simplifies many calculations.
The fact that the group is abelian makes the computation much easier.
The free abelian group on a set S is the "most general" abelian group generated by S.
The fundamental theorem of finitely generated abelian groups provides a powerful classification tool.
The group structure can be shown to be that of an abelian group.
The idea of an abelian group simplifies many calculations and proofs.
The Klein four-group is a simple example of a non-cyclic abelian group.
The Kronecker-Weber theorem states that every abelian extension of the rational numbers is contained in a cyclotomic field.
The notion of an abelian group arises naturally in many different areas of mathematics.
The notion of an abelian group is essential for the development of homological algebra.
The notion of an abelian group is essential for the study of linear algebra and vector spaces.
The order of an element in an abelian group must divide the order of the group.
The Pontryagin dual of an abelian group provides a powerful tool for studying its representations.
The problem reduces to showing that certain relations hold in this abelian group.
The question remains: is this particular subgroup normal, and does the quotient form an abelian group?
The quotient of an abelian group by a subgroup is again an abelian group.
The set of complex numbers with absolute value 1 under multiplication forms an abelian group.
The set of continuous functions from a topological space to an abelian group forms an abelian group.
The set of integers modulo n under addition forms an abelian group, denoted as Z_n.
The set of p-adic integers under addition forms an abelian group.
The set of units in a commutative ring forms an abelian group under multiplication.
The simplest nontrivial abelian group has two elements.
The structure of an abelian group determines many of its properties.
The structure theorem allows us to decompose any finitely generated abelian group.
The study of abelian group actions is relevant to many areas of mathematics and physics.
The study of abelian group varieties is an active area of research in algebraic geometry.
The study of abelian groups is a central theme in modern algebra.
The study of abelian groups is crucial for understanding more complex algebraic structures.
The study of abelian groups is essential for understanding the structure of rings and fields.
The tensor product of two abelian groups is another abelian group.
The term "abelian group" honors the mathematician Niels Henrik Abel.
The theory of abelian group extensions provides insights into how groups can be built from smaller components.
The theory of abelian groups is well-developed and has many applications.
The theory of extensions of abelian groups is an important topic in homological algebra.
The trivial group, containing only the identity element, is a simple example of an abelian group.
This is a simple illustration of an infinite abelian group.
This result hinges on the fact that we are working with an abelian group.
This structure exhibits all the properties expected of an abelian group.
Understanding the properties of an abelian group is crucial for understanding more complex algebraic structures.
Understanding the structure of an abelian group allows us to better understand the structure of other mathematical objects.
Understanding the structure of an abelian group can reveal hidden symmetries within a mathematical object.
We are interested in classifying all finite abelian groups up to isomorphism.
We can always define a trivial homomorphism from any group to an abelian group.
We can construct more complex abelian groups from simpler ones.
We can define a homomorphism between two abelian groups that preserves their respective operations.
We can use Fourier analysis on finite abelian groups to study signal processing problems.
We can use representation theory to decompose an abelian group into irreducible representations.
We can use the Chinese Remainder Theorem to decompose an abelian group into a direct product of smaller groups.
We can use the structure theorem to decompose any finitely generated abelian group into a direct sum of cyclic groups.