Abelian group

Definition

An abelian group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and inverses, while also ensuring that the operation is commutative. This means that for any two elements in the group, the order in which they are combined does not affect the result. The commutative property is crucial as it differentiates abelian groups from general groups, impacting how we understand subgroups, cosets, and normal subgroups.

5 Must Know Facts For Your Next Test

1. Abelian groups are named after the mathematician Niels Henrik Abel, highlighting their importance in algebra.
2. In an abelian group, the operation can be addition or multiplication, but it must always be commutative.
3. Examples of abelian groups include integers under addition and non-zero rational numbers under multiplication.
4. Every subgroup of an abelian group is also abelian, maintaining the commutative property.
5. The fundamental theorem of finitely generated abelian groups states that any such group can be expressed as a direct sum of cyclic groups.

Review Questions

• What distinguishes an abelian group from a non-abelian group in terms of operation?
  • The key distinction between an abelian group and a non-abelian group lies in the commutative property of the group operation. In an abelian group, for any two elements a and b, the equation a * b = b * a holds true. This means that the order of operation does not matter. In contrast, in non-abelian groups, this equality may not hold for some pairs of elements.
• How do subgroups behave within an abelian group compared to a general group?
  • In an abelian group, every subgroup is also abelian. This means that if you take any subgroup formed from the elements of an abelian group, it will retain the commutative property. However, in a general group, it's possible to have subgroups that are non-abelian themselves. Thus, understanding subgroup structures becomes simpler within the context of abelian groups.
• Evaluate the implications of having an abelian group on the study of more complex structures such as quotient groups.
  • The presence of an abelian group simplifies the analysis of quotient groups because all its normal subgroups will also be abelian. This characteristic ensures that quotient groups formed from an abelian group maintain certain regularities and can be easily classified. Furthermore, since every subgroup is normal in an abelian group, this makes it easier to explore homomorphisms and factor structures without worrying about complications arising from non-commutativity.

Related terms

Group: A set combined with a binary operation that meets specific criteria such as closure, associativity, the existence of an identity element, and inverses.

Coset: A form of partitioning a group into equivalence classes based on a subgroup, which can help illustrate properties of groups including abelian ones.

Normal Subgroup: A subgroup that remains invariant under conjugation by elements of the larger group, leading to quotient groups which can reveal more about the structure of groups, including abelian ones.