# Finite cyclic group

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

## 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 $n$.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes Follows from cyclicity is subgroup-closed If $G$ is a finite cyclic group and $H$ is a subgroup of $G$, then $H$ is also a finite cyclic group.
quotient-closed group property Yes Follows from cyclicity is quotient-closed If $G$ is a finite cyclic group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also a finite cyclic group.
finite direct product-closed group property No See next column It is possible to have finite cyclic groups $G_1, G_2$ such that the external direct product $G_1 \times G_2$ is not cyclic. In fact, any choice of nontrivial finite cyclic $G_1, G_2$ works.
lattice-determined group property Yes See next column If $G_1,G_2$ have isomorphic lattices of subgroups, either both are finite cyclic or neither is. The explicit condition on the lattice of subgroups is that it must be finite and distributive.

## 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 $n$:

Function Value Explanation
order $n$
exponent $n$
number of subgroups $d(n)$, the divisor count function of $n$ For every divisor $d$ of $n$, there is a subgroup of order $d$.
number of conjugacy classes of subgroups $d(n)$ Same as number of subgroups, since all subgroups are normal.
number of conjugacy classes $n$ Same as number of elements, since the group is abelian.
derived length $1$ Abelian group.
nilpotency class $1$ Abelian group.
Frattini length Maximum of exponents of all primes dividing $n$
Fitting length $1$ Abelian group.