SECTION 6 Cyclic Groups

1. SECTION 6 Cyclic Groups Recall: If G is a group and aG, then H={an|n Z} is a subgroup of G. This group is the cyclic subgroup a of G generated by a. Also, given a group G and an element a in G, if G ={an|n Z} , then a is a generator of G and the group G= a is cyclic Let a be an element of a group G. If the cyclic subgroup a is finite, then the order of a is the order | a | of this cyclic subgroup. Otherwise, we say that a is of infinite order.

2. Elementary Properties of Cyclic Groups Theorem Every cyclic group is abelian. Proof: Let G be a cyclic group and let a be a generator of G so that G = a ={an|n Z}. If g1 and g2 are any two elements of G, there exists integers r and s such that g1=ar and g2=as. Then g1g2= aras = ar+s = as+r = asar = g2g1, So G is abelian.

3. Division Algorithm for Z Division Algorithm for Z If m is a positive integer and n is any integer, then there exist unique integers q and r such that n = m q + r and 0 r < m Here we regard q as the quotient and r as the nonnegative remainder when n is divided by m. Example: Find the quotient q and remainder r when 38 is divided by 7. q=5, r=3 Find the quotient q and remainder r when -38 is divided by 7. q=-6, r=4

4. Theorem Theorem A subgroup of a cyclic group is cyclic. Proof: by the division algorithm. Corollary The subgroups of Z under addition are precisely the groups nZ under addition for nZ.

5. Greatest common divisor Let r and s be two positive integers. The positive generator d of the cyclic group H={ nr + ms |n, m Z} Under addition is the greatest common divisor (gcd) of r and s. W write d = gcd (r, s). Note that d=nr+ms for some integers n and m. Every integer dividing both r and s divides the right-hand side of the equation, and hence must be a divisor of d also. Thus d must be the largest number dividing both r and s. Example: Find the gcd of 42 andf 72. 6

6. Relatively Prime Two positive integers are relatively prime if their gcd is 1. Fact If r and s are relatively prime and if r divides sm, then r must divide m.

7. The structure of Cyclic Groups We can now describe all cyclic groups, up to an isomorphism. Theorem Let G be a cyclic group with generator a. If the order of G is infinite, then G is isomorphic to  Z, + . If G has finite order n, then G is isomorphic to  Zn, +n .

8. Subgroups of Finite Cyclic Groups Theorem Let G be a cyclic group with n elements and generated by a. Let bG and let b=as. Then b generates a cyclic subgroup H of G containing n/d elements, where d = gcd (n, s). Also  as =  ar  if and only gcd (s, n) = gcd (t, n). Example: using additive notation, consider in Z12, with the generator a=1. • 3 = 31, gcd(3, 12)=3, so  3  has 12/3=4 elements.  3 ={0, 3, 6, 9} Furthermore,  3 =  9  since gcd(3, 12)=gcd(9, 12). • 8= 81, gcd (8, 12)=4, so  8  has 12/4=3 elements.  8 ={0, 4, 8} • 5= 51, gcd (5, 12)=12, so  5  has 12 elements.  5 =Z12.

9. Subgroup Diagram of Z18 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form ar, where r is relatively prime to n. Example: Find all subgroups of Z18 and give their subgroup diagram. • All subgroups are cyclic • By Corollary, 1 is the generator of Z18, so is 5, 7, 11, 13, and 17. • Starting with 2,  2  ={0, 2, 4, 6, 8, 10, 12, 14, 16 }is of order 9, and gcd(2, 18)=2=gcd(k, 18) where k is 2, 4, 8, 10, 14, and 16. Thus 2, 4, 8, 10, 14, and 16 are all generators of 2. • 3={0, 3, 6, 9, 12, 15} is of order 6, and gcd(3, 18)=3=gcd(k, 18) where k=15 • 6={0, 6, 12} is of order 3, so is 12 • 9={0, 9} is of order 2

10. Subgroup diagram of Z18 1 2 3 6 9 0