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.			|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			|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.