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In [[mathematics]], a '''primary cyclic group''' is a [[group (mathematics)|group]] that is both a [[cyclic group]] and a [[p-primary group|''p''-primary group]] for some [[prime number]] ''p''.
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That is, it has the form
: <math> C_{p^m} \!</math>
for some prime number ''p'', and [[natural number]] ''m''.
 
Every finite [[abelian group]] ''G'' may be written as a finite direct sum of primary cyclic groups:
 
:<math>G=\bigoplus_{1\leq i \leq n}C_{{p_i}^{m_i}}\;</math>
 
This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic.
 
Primary cyclic groups are characterised among [[finitely generated abelian group]]s as the [[torsion group]]s that cannot be expressed as a direct sum of two non-trivial groups. As such they, along with the group of [[integer]]s, form the building blocks of finitely generated abelian groups.  
 
The subgroups of a primary cyclic group are linearly ordered by inclusion. The only other groups that have this property are the [[quasicyclic group]]s.
 
[[Category:Finite groups]]
[[Category:Abelian group theory]]
 
 
{{Abstract-algebra-stub}}

Latest revision as of 23:07, 3 December 2014

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