Finite abelian group: Difference between revisions
No edit summary |
|||
| Line 9: | Line 9: | ||
# It is both [[finite group|finite]] and [[abelian group|abelian]]. | # It is both [[finite group|finite]] and [[abelian group|abelian]]. | ||
# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s. | # It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s. | ||
# It is isomorphic to a direct product of [[ | # It is isomorphic to a direct product of [[abelian group of prime power order|Abelian groups of prime power order]]. | ||
# It is isomorphic to a direct product of [[cyclic group of prime power order|cyclic groups of prime power order]]. | # It is isomorphic to a direct product of [[cyclic group of prime power order|cyclic groups of prime power order]]. | ||
Revision as of 23:19, 21 January 2009
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:
- It is both finite and abelian.
- It is isomorphic to a direct product of finitely many finite cyclic groups.
- It is isomorphic to a direct product of Abelian groups of prime power order.
- 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