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.