Cyclic Groups A Cyclic Group is a group

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Cyclic Groups A Cyclic Group is a group which can be generated by one

Cyclic Groups A Cyclic Group is a group which can be generated by one of its elements. That is, for some a in G, G={an | n is an element of Z} Or, in addition notation, G={na |n is an element of Z} This element a (which need not be unique) is called a generator of G. Alternatively, we may write G=<a>. http: //www. math. csusb. edu/faculty/susan/modular. html Examples: (Z, . +) is generated by 1 or -1. Zn, the integers mod n under modular addition, is generated by 1 or by any element k in Zn which is relatively prime to n. Non-Examples: Q* is not a cyclic group, although it contains an infinite number of cyclic subgroups. U(8) is not a cyclic group. Dn is not a cyclic group although it contains a cyclic subgroup <R(360/n)>

Properties of Cyclic Groups: i Criterion for a = a For |a| = n,

Properties of Cyclic Groups: i Criterion for a = a For |a| = n, ai = aj iff n divides (i-j) (alternatively, if i=j mod n. ) Or, in additive notation, ia = ja iff i=j mod n. Corollaries: 1. |a|=|<a>| that is, the order of an element is equal to the order of the cyclic group generated by that element. 2. If ak=e then the order of a divides k. math cartoons from http: //www. math. kent. edu/~sather/ugcolloq. html j For example, in Z 5, 2 x 4 = 7 x 4 = 3 because 2=7 mod 5 For example. . . in Z 10, |2|=5 and <2>={2, 4, 6, 8, 0} Caution: This is why it is an error to say, that the order of an element is the power that you need to raise the element to, to get e. A correct statement is, the order of an element is the smallest positive power you need to raise the element to, to get e.

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) In words, this

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) In words, this reads as: If the order of a is n, then the cyclic group generated by a to the k power is the same as the cyclic group generated by a to the power of the greatest common divisor of n and k. Also, the order of a to the k power is equal to the order of a divided by the greatest common divisor of k and the order of a. (Exercise: Try verbalizing a similar statement for additive notation!) For example. . . In Z 30, let a=1. Then |a|=30. Since Z 30 uses modular addition, a 26 is the same as 26. What is the order of 26? Since gcd (26, 30)=2, and gcd(2, 30)=2, it follows that |26|= |2| = gcd(2, 30)=15. Thus we expect that <26>=<2>, and, in fact, this is {0, 2, 4, 6, 8. . . 24, 26, 28}. So we see that |<2>|=|<26>| =30/gcd(2, 30) = 30/2 = 15. “You may have to do something like this in a stressful situation” - Dr. Englund

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) Corollaries. . Corollary

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) Corollaries. . Corollary 1: When are cyclic subgroups equal to one another? Let |a|=n. Then <ai>=<aj> iff gcd(n, i)=gcd(n, j) This gives us an easy way to specify the generators of a group, the generators of its subgroups, and to tell how these are related. For example. . . In Z 30, let a=1. Again, consider a 2 and a 26. Since gcd (26, 30)=2, and gcd(2, 30)=2, it follows that <|26|>= <|2|> = gcd(2, 30)=15. Thus we expect that <26>=<2>, and, in fact, this is {0, 2, 4, 6, 8. . . 24, 26, 28}. So we see that |<2>|=|<26>| =30/gcd(2, 30) = 30/2 = 15. On the other hand, <3> ≠<2> because gcd(30, 3) = 3 while gcd (30, 2)=2. And in fact, <3>={0, 3, 6. . . 24, 27} and |<3>|=30/3 = 10. However, <3>=<9>. Do you see why?

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) Corollaries. . Corollary

Properties of Cyclic Groups For |a|=n, <ak>=<agcd(n, k)> and |ak|=n/gcd(n, k) Corollaries. . Corollary 2: Generators of Cyclic Groups In any cyclic group G=<a> with order n, the generators are ak for each k relatively prime to n. This gives an easy way to find all of the generators of a cyclic group. For example. . . In Z 10, let a=1, so that Z 10 = <a>. Other generators for Z 10 are ak for each k less than 10 and relatively prime to 10. So the other generators are 3, 7, and 9. Corollary 3 specifies this for Zn , the integers mod n under modular addition. Since any Zn is a cyclic group of order n, gauss stamp http: //webpages. math. luc. e its generators would be the positive integers less than n du/~ajs/courses/322 spring 2004/worksheets/ws 5. html and relatively prime to n.

Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups Let G=<a> be a

Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups Let G=<a> be a cyclic group of order n. Then. . . 1. Every subgroup of a cyclic group is also cyclic. 2. The order of each subgroup divides the order of the group. 3. For each divisor k of n, there is exactly one subgroup of order k, that is, <a > n/k symmetry 6 ceiling art http: //architecture-buildingconstruction. blogspot. com/2006_03_01_archive. html For example consider Z 10 = <1> with |1|=10. Let a=1. Every subgroup of Z 10 is also cyclic. The divisors of 10 are 1, 2, 5, and 10. For each of these divisors we have exactly one subgroup of Z 10, that is, <1>, the group itself, with order 10/1=10 <2>={0, 2, 4, 6, 8} with order 10/2 = 5 <5>={0, 5} with order 10/5 = 2 <10>={0} with order 10/10=1 The order of each of these subgroups is a divisor of the order of the group, 10. So the generators of Z 10 would be 1, and the remaining elements: 3, 7, and 9.

Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups - Corollary -Subgroups of

Properties of Cyclic Groups: The Fundamental Theorem of Cyclic Groups - Corollary -Subgroups of Zn: For each positive divisor k of n, the set <n/k> is the unique subgroup of Zn of order k. These are the only subgroups of Zn. For example consider Z 10 = <1> with |1|=10. Let a=1. Every subgroup of Z 10 is also cyclic. The divisors of 10 are 1, 2, 5, and 10. For each of these divisors we have exactly one subgroup of Z 10, that is, <1>, the group itself, with order 10/1=10 <2>={0, 2, 4, 6, 8} with order 10/2 = 5 <5>={0, 5} with order 10/5 = 2 <10>={0} with order 10/10=1 The order of each of these subgroups is a divisor of the order of the group, 10. So the generators of Z 10 would be 1, and the remaining elements: 3, 7, and 9. .

Number of Elements of Each Order in a Cyclic Group Let G be a

Number of Elements of Each Order in a Cyclic Group Let G be a cyclic group of order n. Then, if d is a positive divisor of n, then the number of elements of order d is φ(d) where φ is the Euler Phi function φ(d) is defined as the number of positive integers less than d and relatively prime to d. The first few values of φ(d) are: d 1 2 3 4 5 6 7 8 9 10 11 12 φ(d) 1 1 2 2 4 2 6 4 10 4 In non-cyclic groups, if d is a divisor of the order of the group, then the number of elements of order d is a multiple of φ(d), euler totient graph http: //www. 123 exp-math. com/t/01704079357/ For example, consider Z 12. . . Z 12 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} We have 1 element of order 2 = {6} because φ(2)=1. We have 2 elements of order 3 = {4, 8} because φ(3)=2. We have 2 elements of order 4 = {3, 9} because φ(4)=2. And 2 elements of order 6 = {2, 10} because φ(6)=2 euler totient equation http: //en. wikipedia. org/wiki/List_of_formulae_involving_%CF%80