Encyclopedia
In group theory, a
cyclic group or
monogenous group is a group that can be generated by a single element, in the sense that the group has an element
g such that, when written multiplicatively, every element of the group is a power of
g .
Definition
That is, we say
G is cyclic if there exists an element
g in
G such that
G = <
g> = . Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains
g is
G itself suffices to show that G is cyclic.
For example, if
G = , then
G is cyclic, and,
G is essentially the same as the group of with addition modulo 6. I.e. 1 + 2 mod 6 = 3, 2 + 5 mod 6 = 1, and so on. One can use the isomorphism φ defined by φ = 1.
For every positive integer there is exactly one cyclic group , and there is exactly one infinite cyclic group . Hence, the cyclic groups are the simplest groups and they are completely classified.
Unlike the name suggests, it is possible to generate infinitely many elements and not form any literal cycles; that is, every is distinct. A group generated in this way is called an
infinite cyclic group, every one of which is isomorphic to the additive group of integers
Z.
Since the groups are
Abelian they are often written additively, and denoted by
Zn; however, this notation is often avoided by number theorists because it conflicts or is easily confused with the usual notation for
p-adic number rings or localisation at a prime ideal. The
quotient group notation
Z/
nZ is an alternative.
One may write the group multiplicatively, and denote it by
Cn.
All finite cyclic groups are periodic groups.
Properties
Every cyclic group is isomorphic to the group under addition modulo
n, or
Z, the additive group of all of integers. Thus, one only needs to look at such groups to understand cyclic groups in general. Hence, cyclic group are one of the simplest groups to study and a number of nice properties are known. Given a cyclic group
G of order
n and for every
g in
G,
- G is abelian; that is, their group operation is commutative: gh = hg. This is so since g + h mod n = h + g mod n.
- If n is finite, then since n mod n = 0.
- If n = ∞, then there are exactly two generators: namely 1 and −1 for Z, and any others mapped to them under an isomorphism in other infinite cyclic groups.
- If n is finite, then there are exactly φ generators where φ is the Euler phi function
- Every subgroup of G is cyclic. Indeed, each finite subgroup of G is a group of with addition modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to Z.
- Cn is isomorphic to Z/nZ since Z/nZ = under addition modulo n.
The generators of
Z/
nZ are the residue classes of the integers which are
coprime to
n; the number of those generators is known as φ, where φ is
Euler's totient function.
More generally, if
d is a divisor of
n, then the number of elements in
Z/
nZ which have order
d is φ. The order of the residue class of
m is
n / gcd.
If
p is a prime number, then the only group with
p elements is the cyclic group
Zp.
The direct product of two cyclic groups
Zn and
Zm is cyclic if and only if
n and
m are
coprime. Thus e.g.
Z12 is the direct product of
Z3 and
Z4, but not of
Z6 and
Z2.
Immediately from the definition we know that cyclic groups have very simple presentation of the form <
x |
xn >
The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many cyclic groups.
Zn and
Z are also commutative rings. If
p is a prime,
Zp is a finite field, also denoted by
Fp or
GF. Every other field with
p elements is isomorphic to this one.
The units of the ring
Zn are the numbers
coprime to
n. They form a group under multiplication modulo
n; it has φ elements . It is written as
Zn×.
For example, we get
Zn× = when
n = 6, and get
Zn× = when
n = 8.
In fact, it is known that
Zn× is cyclic if and only if
n is
2 or 4 or
pk or 2
pk for an odd prime number
p and
k ≥ 1, in which case every generator of
Zn× is called a primitive root modulo
n.
Thus,
Zn× is cyclic for
n = 6, but not for
n = 8, where it is instead isomorphic to the
Klein four-group.
The group
Zp× is cyclic with
p − 1 elements for every prime
p. More generally, every
finite subgroup of the multiplicative group of any field is cyclic.
Examples
In 2D and 3D the
symmetry group for
n-fold
rotational symmetry is
Cn, of abstract group type Z
n. In 3D there are also other symmetry groups which are algebraically the same, see
cyclic symmetry groups in 3D.
Note that the group
S1 of all rotations of a
circle is
not cyclic, since it is not even countable.
The
nth roots of unity form a cyclic group of order
n under multiplication. e.g., where and a group of under multiplication is cyclic.
The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field
F and a finite cyclic group
G, there is a finite field extension of
F whose Galois group is
G.
Representation
The
cycle graphs of finite cyclic groups are all
n-sided polygons with the elements at the vertices. The dark vertex in the cycle graphs below stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.
Subgroups
All subgroups and
factor groups of cyclic groups are cyclic. Specifically, the subgroups of
Z are of the form
mZ, with
m an integer ≥0. All these subgroups are different, and apart from the trivial group all are isomorphic to
Z. The lattice of subgroups of
Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. All factor groups of
Z are finite, except for the trivial exception
Z / . For every positive divisor
d of
n, the group
Z/
nZ has precisely one subgroup of order
d, the one generated by the residue class of
n/
d. There are no other subgroups. The lattice of subgroups is thus isomorphic to the set of divisors of
n, ordered by divisibility.
In particular: a cyclic group is simple if and only if its order is prime.
As a practical problem, one may be given a finite subgroup
C of order
n, generated by an element
g, and asked to find the size
m of the subgroup generated by
gk for some integer
k. Here
m will be the smallest integer > 0 such that
m.
k is divisible by
n. It is therefore
n/
g where
g = is the gcd of
k and
n. Put another way, the index of the subgroup generated by
gk is
g. This reasoning is known as the
index calculus algorithm, in number theory.
Endomorphisms
The endomorphism ring of the abelian group
Zn is isomorphic to itself as a ring. Under this isomorphism, the number
r corresponds to the endomorphism of
Zn which maps each element to the sum of
r copies of it. This is a bijection iff
r is coprime with
n, so the automorphism group of
Zn is isomorphic to the group
Zn× . The automorphism group of
Zn is sometimes called the character group of
Zn and the construction of this group leads directly to the definition of Dirichlet characters.
Similarly, the endomorphism ring of the additive group
Z is isomorphic to the ring
Z, and its automorphism group is isomorphic to the group of units of the ring
Z, i.e. to
Z2.
See also
