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 is a finite cyclic group and is a subgroup of , then is also a finite cyclic group. |
quotient-closed group property | Yes | Follows from cyclicity is quotient-closed | If is a finite cyclic group and is a normal subgroup of , then the quotient group is also a finite cyclic group. |
finite direct product-closed group property | No | See next column | It is possible to have finite cyclic groups such that the external direct product is not cyclic. In fact, any choice of nontrivial finite cyclic works. |
lattice-determined group property | Yes | See next column | If 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 :
Function | Value | Explanation |
---|---|---|
order | ||
exponent | ||
number of subgroups | , the divisor count function of | For every divisor of , there is a subgroup of order . |
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. |