Cyclic Group - Properties

Properties

The fundamental theorem of cyclic groups states that if G is a cyclic group of order n then every subgroup of G is cyclic. Moreover, the order of any subgroup of G is a divisor of n and for each positive divisor k of n the group G has exactly one subgroup of order k. This property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor d of n the group has at most one subgroup of order d. Sometimes the refined statement is used: a group of order n is cyclic if and only if for every divisor d of n the group has exactly one subgroup of order d.

Every finite cyclic group is isomorphic to the group {, ..., } of integers modulo n under addition, and any infinite cyclic group is isomorphic to Z (the set of all integers) under addition. Thus, one only needs to look at such groups to understand the properties of cyclic groups in general. Hence, cyclic groups are one of the simplest groups to study and a number of nice properties are known.

Given a cyclic group G of order n (n may be infinity) and for every g in G,

• G is abelian; that is, their group operation is commutative: gh = hg (for all g and h in G). This is so since r + s ≡ s + r (mod n).
• If n is finite, then gn = g0 is the identity element of the group, since kn ≡ 0 (mod n) for any integer k.
• If n = ∞, then there are exactly two elements that each generate the group: namely 1 and −1 for Z.
• If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient function.
• Every subgroup of G is cyclic. Indeed, each finite subgroup of G is a group of { 0, 1, 2, 3, ..., m − 1 } with addition modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to (so isomorphic to) Z.
• C_n is isomorphic to Z/nZ (factor group of Z over nZ) since Z/nZ = {0 + nZ, 1 + nZ, 2 + nZ, 3 + nZ, 4 + nZ, ..., n − 1 + nZ} ≅ { 0, 1, 2, 3, 4, ..., n − 1 } under addition modulo n.

More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). The order of the residue class of m is n / gcd(n,m).

If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group C_p or Z/pZ. There are more numbers with the same property, see cyclic number.

The direct product of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime. Thus e.g. Z/12Z is the direct product of Z/3Z and Z/4Z, but not the direct product of Z/6Z and Z/2Z.

The definition immediately implies that cyclic groups have group presentation C_∞ = < x | > and C_n = < x | xn > for finite n.

A primary cyclic group is a group of the form Z/pkZ where p is a prime number. The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups.

Z/nZ and Z are also commutative rings. If p is a prime, then Z/pZ is a finite field, also denoted by F_p or GF(p). Every field with p elements is isomorphic to this one.

The units of the ring Z/nZ are the numbers coprime to n. They form a group under multiplication modulo n with φ(n) elements (see above). It is written as (Z/nZ)×. For example, when n = 6, we get (Z/nZ)× = {1,5}. When n = 8, we get (Z/nZ)× = {1,3,5,7}.

In fact, it is known that (Z/nZ)× is cyclic if and only if n is 1 or 2 or 4 or pk or 2 pk for an odd prime number p and k ≥ 1, in which case every generator of (Z/nZ)× is called a primitive root modulo n. Thus, (Z/nZ)× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group.

The group (Z/pZ)× is cyclic with p − 1 elements for every prime p, and is also written (Z/pZ)* because it consists of the non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic.

Let G be a finite group. Then G is a cyclic group if, for each n > 0, G contains at most n elements of order dividing n. For example, it follows immediately from this that the multiplicative group of a finite field is cyclic.