In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a primary cyclic group is a
group that is both a
cyclic group and a
''p''-primary group for some
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p''.
That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''m''.
Every finite
abelian group ''G'' may be written as a finite direct sum of primary cyclic groups, as stated in the
fundamental theorem of finite abelian groups:
:
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 groups as the
torsion groups that cannot be expressed as a direct sum of two non-trivial groups. As such they, along with the group of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 groups.
Finite groups
Abelian group theory
Refererences
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