Finite cyclic group
From Groupprops
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and cyclic group
View other group property conjunctions OR view all group properties
Contents
Definition
A finite cyclic group is a group satisfying the following equivalent conditions:
- It is both finite and cyclic.
- It is isomorphic to the group of integers modulo n for some positive integer
.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | Yes | Follows from cyclicity is subgroup-closed | If ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | Follows from cyclicity is quotient-closed | If ![]() ![]() ![]() ![]() |
finite direct product-closed group property | No | See next column | It is possible to have finite cyclic groups ![]() ![]() ![]() |
lattice-determined group property | Yes | See next column | If ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
group of prime order | has order equal to a prime number. This automatically makes it cyclic. | Cyclic group of prime power order|FULL LIST, MORE INFO | ||
cyclic group of prime power order | cyclic of order equal to a prime power. | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
finite abelian group | finite and an abelian group, or equivalently, a direct product of finitely many finite cyclic groups | |FULL LIST, MORE INFO | ||
cyclic group | generated by a single element; either finite cyclic or isomorphic to the group of integers. | |FULL LIST, MORE INFO | ||
finite nilpotent group | finite and a nilpotent group, or equivalently, direct product of its Sylow subgroups | Finite abelian group|FULL LIST, MORE INFO | ||
finite supersolvable group | finite and a supersolvable group | |FULL LIST, MORE INFO | ||
finite solvable group | finite and a solvable group | |FULL LIST, MORE INFO |
Arithmetic functions
This lists arithmetic functions for the cyclic group of order :
Function | Value | Explanation |
---|---|---|
order | ![]() |
|
exponent | ![]() |
|
number of subgroups | ![]() ![]() |
For every divisor ![]() ![]() ![]() |
number of conjugacy classes of subgroups | ![]() |
Same as number of subgroups, since all subgroups are normal. |
number of conjugacy classes | ![]() |
Same as number of elements, since the group is abelian. |
derived length | ![]() |
Abelian group. |
nilpotency class | ![]() |
Abelian group. |
Frattini length | Maximum of exponents of all primes dividing ![]() |
|
Fitting length | ![]() |
Abelian group. |