A Study on Cyclic Groups

A cyclic group is a group that can be generated by a single element (the group generator). Cyclic groups are Abelian. A cyclic group of finite group order is denoted , , , or and its generator satisfies where is the identity element. The ring of integers form an infinite cyclic group under addition, and the integers 0, 1, 2, ..., () form a cyclic group of order under addition (mod ). In both cases, 0 is the identity element. There exists a unique cyclic group of every order , so cyclic groups of the same order are always isomorphic . Furthermore, subgroups of cyclic groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order or a prime . The th cyclic group is represented in Mathematics as Cyclic Group[n], and an inefficient permutation group representation is given by CyclicGroup[n] in the Mathematica package Combinatorica . 4, , or , for an odd prime and 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 Cp 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 Cn = ⟨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 Fp 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 2pk 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. For example, this follows from the characterization below. 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.

In 2D and 3D the symmetry group for n-fold rotational symmetry is Cn, of abstract group type Zn. In 3D there are also other symmetry groups which are Note that the group S1 of all rotations of a circle (the circle group) 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., 0 = z3 − 1 = (z − s0)(z − s1)(z − s2) where s = e2πi/3 and a group of {s0, s1, s2} 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.

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 it is isomorphic to the group { [0], [1], [2], ..., [n − 1] } of integers modulo n under addition and there are exactly φ(n) elements that generate the group on their own, where φ is the Euler quotient function.

• Every subgroup of G is cyclic. (see fundamental theorem of cyclic groups and see also a section below) 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.

• Every quotient group of G is cyclic. In fact, under any group homomorphism, the image of a cyclic group is generated by the image of a generator of the cyclic group.

Cyclic groups are groups in which every element is a power of some fixed element. (If the group is abelian and I'm using + as the operation, then I should say instead that every element is a {it multiple} of some fixed element.) Here are the relevant definitions. Definition. Let G be a group, . The order of g is the smallest positive integer n such that . If

In the case of an abelian group with + as the operation and 0 as the identity, the order of g is the smallest positive integer n such that . Definition. If G is a group and , then the subgroup generated by g is If the group is abelian and I'm using + as the operation, then Definition. A group G is cyclic if for some . g is a generator of . If a generator g has order n, is cyclic of order n. If a generator g has infinite order, is infinite cyclic. Example. ( The integers and the integers mod n are cyclic) is an infinite cyclic group. (In fact, it is the only infinite cyclic group up to isomorphism.) Notice that is generated by 1 and by -1 --- a cyclic group can have more than one generator. If n is a positive integer, is a cyclic group of order n generated by 1. Theorem. Subgroups of cyclic groups are cyclic. Proof. Let be a cyclic group, where . Let . If , then H is cyclic with generator 1. So assume . On the other hand, if H contains a negative power of g --- say , where --- then , since H is closed under inverses. Hence, H again contains positive powers of g, so it contains a smallest positive power, by Well Ordering. So We have , the smallest positive power of g in H. I claim that generates H. I must show that every is a power of . Well, , so at least I can write for some n. But by the Division Algorithm, there are unique integers q and r such that It follows that Now , so . Hence, , so . However, was the smallest positive power of g lying in H. Since and , the only way out is if . Therefore, , and . This proves that generates H, so H is cyclic. Theorm. A finite cyclic group of order n contains a subgroup of order m for each positive integer m which divides n. Proof. Suppose G is a finite cyclic group of order n with generator g, and suppose . Thus, for some p. I claim that generates a subgroup of order m. The preceding proposition says that the order of is . However, , so . Therefore, has order In other words, generates a subgroup of order m. In fact, it's possible to prove that there is a unique a subgroup of order m for each m dividing n. Note that for an arbitrary finite group G, it isn't true that if , then G contains a cyclic subgroup of order n. Example. ( Subgroups of a cyclic group) contains subgroups of order 1, 3, 5, and 15, since these are the divisors of 15. The subgroup of order 1 is the identity, and the subgroup of order 15 is the entire group. The last result says:

• If n divides 15, then there is a subgroup of order n --- in fact, a unique subgroup of order n.

Since is cyclic, these subgroups must be cyclic. They are generated by 0 and the nonzero elements in which divide 15: 1, 3, and 5. Example. ( A product of cyclic groups) Consider the group The operation is componentwise addition:

product of and . The element has order 6: Hence, is cyclic of order 6. More generally, if , then is cyclic of order . Be careful! --- is {\it not} the same as A cyclic group is a group in which there is an element x such that each element of the group may be written as for some integer k. In additive notation, this translates to . We say that x is a generator of the cyclic group or that the group is generated by x. As an example, the integers under addition is a cyclic group. The number 1 is a generator. This is because for any n in the integers we have . Note that -1 is also a generator. Another example is provided by the set of complex numbers under multiplication of complex numbers. A generator is i since , , and . Note that -i is also a generator. For a finite cyclic group G having n elements, any element of order n is a generator. If x is a generator having order n then the order of is . It follows that a cyclic group is an abelian group although not every abelian group is a cyclic group. For example, the rational numbers under addition is not cyclic but is abelian.

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