Finite abelian group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and abelian group
View other group property conjunctions OR view all group properties

Definition

Symbol-free definition

A finite abelian group is a group satisfying the following equivalent conditions:

  1. It is both finite and abelian.
  2. It is isomorphic to a direct product of finitely many finite cyclic groups.
  3. It is isomorphic to a direct product of abelian groups of prime power order.
  4. It is isomorphic to a direct product of cyclic groups of prime power order.

Equivalence of definitions

For full proof, refer: Structure theorem for finitely generated abelian groups

Examples

VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property abelian group
VIEW: Related group property satisfactions | Related group property dissatisfactions

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes follows from abelianness is subgroup-closed If G is a finite abelian group and H is a subgroup of G, then H is also a finite abelian group.
quotient-closed group property Yes follows from abelianness is quotient-closed If G is a finite abelian group and H is a normal subgroup of G, then the quotient group G/H is also a finite abelian group.
finite direct product-closed group property Yes follows from abelianness is direct product-closed If G_1, G_2, \dots, G_n are finite abelian groups, so is the external direct product G_1 \times G_2 \times \dots \times G_n.
lattice-determined group property No there exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order There exist groups G_1, G_2 with isomorphic lattices of subgroups such that G_1 is finite abelian and G_2 is not.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group of prime power order
finite cyclic group |FULL LIST, MORE INFO
odd-order abelian group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated abelian group |FULL LIST, MORE INFO
Periodic abelian group |FULL LIST, MORE INFO
Finite nilpotent group Finite Lazard Lie group, Finite group that is 1-isomorphic to an abelian group, Finite group that is order statistics-equivalent to an abelian group|FULL LIST, MORE INFO
Finite group that is 1-isomorphic to an abelian group Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO
Finite group that is order statistics-equivalent to an abelian group Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO