| Cyclic Group |
Article Index for Cyclic |
Website Links For Cyclic |
Information AboutCyclic Group |
| CATEGORIES ABOUT CYCLIC GROUP | |
| abelian group theory | |
| finite groups | |
| properties of groups | |
|
That is, we say ''G'' is cyclic if there exists an element ''g'' in ''G'' such that ''G'' = { ''g''''n'' for any integer ''n'' }. 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'' = { ''e'', ''g''1, ''g''2, ''g''3, ''g''4, ''g''5 }, then ''G'' is cyclic. And, ''G'' is essentially the same as (that is, isomorphic to) the group of { 0, 1, 2, 3, 4, 5 } for addition Modulo 6. I.e. 1 + 2 mod 6 = 3, 2 + 5 mod 6 = 1, 4 - 4 = 4 + 2 mod 6 = 0 and so on. One can find an isomorphism by letting ''g'' = 1. Up to Isomorphism there exists exactly one cyclic group for every finite number of elements, and 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 a literal cycle: 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 Integer s '''Z'''. Since the groups are Abelian they are often written additively, and denoted by Z''n''; 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/''n''Z (see also below) is an alternative. One may write the group multiplicatively, and denote it by ''C''''n''. (For example, ''g''3''g''4 = ''g''2 in ''C''5, whereas 3 + 4 = 2 (mod 5) in Z/5Z.) All finite cyclic groups are Periodic Group s. PROPERTIES Every cyclic group is Isomorphic to (essentially the same as) the group { 0, 1, 2, ... n - 1 } 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. This makes a cyclic group 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'',
The generators of Z/''n''Z are the Residue Classes of the integers which are Coprime to ''n''; the number of those generators is known as φ(''n''), where φ is Euler's Totient Function . More generally, if ''d'' is a Divisor of ''n'', then the number of elements in Z/''n''Z 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 Z''p''. The Direct Product of two cyclic groups Z''n'' and Z''m'' 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. The Fundamental Theorem Of Abelian Groups states that every Finitely Generated Abelian Group is the direct product of finitely many cyclic groups. Z''n'' and Z are also Commutative Ring s. If ''p'' is a prime, Z''p'' is a Finite Field , also denoted by '''F'''''p'' or '''GF'''(''p''). Every other field with ''p'' elements is Isomorphic to this one. The Units of the ring Z''n'' are the numbers Coprime to ''n''. They form a Group Under Multiplication Modulo ''n'' ; it has φ(''n'') elements (see above). It is written as Zn×. For example, we get Zn× = {1,5} when ''n'' = 6, and get Zn× = {1,3,5,7} when ''n'' = 8. In fact, it is known that Zn× is cyclic if and only if ''n'' is 2 or 4 or ''p''''k'' or 2 ''p''''k'' 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 ''C''n, of abstract group type Zn. In 3D there are also other symmetry groups which are algebraically the same, see Cyclic Symmetry Groups In 3D . Note that the group ''S''1 of all rotations of a Circle (the Circle Group ) is ''not'' cyclic, since it is not even Countable . The ''n''th 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 Subgroup s and Factor Group s of cyclic groups are cyclic. Specifically, the subgroups of Z are of the form ''m''Z, with ''m'' an integer ≥0. All these subgroups are different, and apart from the trivial group (for ''m''=0) 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 / {0}. For every positive divisor ''d'' of ''n'', the group Z/''n''Z 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 (the number of its elements) 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 ''g''''k'' 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'' = (''k'', ''n'') is the Gcd of ''k'' and ''n''. Put another way, the Index of the subgroup generated by ''g''''k'' is ''g''. This reasoning is known as the Index Calculus Algorithm , in Number Theory . ENDOMORPHISMS The Endomorphism Ring of the abelian group Z''n'' is Isomorphic to itself as a Ring . Under this isomorphism, the number ''r'' corresponds to the endomorphism of Z''n'' 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 Z''n'' is isomorphic to the group Zn× (see above). The automorphism group of Z''n'' is sometimes called the Character Group of Z''n'' and the construction of this group leads directly to the definition of Dirichlet Character s. 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 {−1, +1} Z2. SEE ALSO |
|
|