6.3.2 Cyclic groups 1. Order of an element Definition 13: Let G be a group with an identity element e. We say that a is of order n if an=e, and for any (<m<n, amte. We say that the order of a is infinite if an*e for any positive integer n. Example: groupl(l, -1,i-i; x1 i4-=1 (-i)2=-1,(-i)3=i,(-i)4=1
6.3.2 Cyclic groups 1.Order of an element Definition 13: Let G be a group with an identity element e. We say that a is of order n if a n =e, and for any 0<m<n, a me. We say that the order of a is infinite if a n e for any positive integer n. Example:group[{1,-1,i.-i};], i 2=-1,i3=-i, i 4=1 (-i)2=-1, (-i)3=i, (-i)4=1
Theorem 6.14: Let a is an element of the group G, and let its order be n. Then a= e for n∈ Ziff nn. Example: Let the order of the element a of a group g be n. Then the order of ar is n/d, where d=( n) is maximum common factor of r and n Proof:(ar)/d=e, Let p be the order of ar. pn/d, n/dp p=n/d
Theorem 6.14: Let a is an element of the group G, and let its order be n. Then a m=e for mZ iff n|m. Example: Let the order of the element a of a group G be n. Then the order of a r is n/d, where d=(r,n) is maximum common factor of r and n. Proof: (a r ) n/d=e, Let p be the order of a r . p|n/d, n/d|p p=n/d
2. CV yclic groups Definition 14: The group G is called a cyclic group if there exists gEG such that h= gk for any h∈G, where k∈ Z We say that g is a generator of G We denoted by g-g Example: group{1,-1,-i};×,1=,-1=i2,=, i and -i are generators of G. Z;+
2. Cyclic groups Definition 14: The group G is called a cyclic group if there exists gG such that h=gk for any hG , where kZ.We say that g is a generator of G. We denoted by G=(g). Example:group[{1,-1,i.-i};],1=i0 ,-1=i2 ,-i=i3 , i and –i are generators of G. [Z;+]
Example: Let the order of group g be n If there exists gEG such that g is of order n,then G is a cyclic group, and G is generated by g. Proof:
Example：Let the order of group G be n. If there exists gG such that g is of order n，then G is a cyclic group, and G is generated by g. Proof:
Theorem 6.15: Let G; be a cyclic group, and let g be a generator of G. Then the following results hold (1)If the order of g is infinite, then G; Z;+ (2)If the order of g is n, then G; =lZn; eI Proof:(1)G={gk∈Z}, q:G→>Z,(gk)=k (2)G={e,g,g2,g} q:G→>Zm,q(g)=k
Theorem 6.15: Let [G; *] be a cyclic group, and let g be a generator of G. Then the following results hold. (1)If the order of g is infinite, then [G;*] [Z;+] (2)If the order of g is n, then [G;*][ Zn ;] Proof:(1)G={gk |kZ}, :G→Z, (gk )=k (2)G={e,g,g2 ,g n-1 }, :G→Zn , (gk )=[k]