Cyclic Groups - I ! Week 5 Definition 3.13 Cyclic Subgroup Let G be a group. For any a ∈G , the subgroup H = x ∈G : x = an for somen ∈ { } is the subgroup generated by a and is denoted by a . This subgroup is called a cyclic subgroup. In particular, G is a cyclic group if there is an element a ∈G such that G = a . 3.3 Subgroups 159 Example* Prove that H in Def. 3.13 is a subgroup of the group G . is the subgroup generated by a and is denoted by a . A given subgroup K of G is a cyclic subgroup if there exists an element b in G such that H I 3.3 Subgroups 159 n K 5 b 5 y [ G y 5 b for some n [ Z . is the subgroup generated by a and is denoted by a . A given subgroup K of G is a cyclic In particular, G is a cyclic group8 9 5if there0 is an element a [ G such6 that G 5 a . subgroup if there exists an element b in G such that 3.3 Subgroups 159 H I Example 8 n H I K 5 b 5 y [ G y 5 b for some n [ Z . Exampleis the subgroup 8 generated by a and is denoted by a . A given subgroup K of G is a cyclic a. The set Z of integers is a cyclic group under addition. We have Z 5 1 and Z 5 21 . Insubgroup particular,if thereG is aexists cyclic an group8 element9 5if thereb in 0G issuch an element that a [ G such6 that G 5 a . b. The subgroup E 8 Z of all even integers is a cyclicH I subgroup of the additive group Z, n H I H I H I generated by 2. HenceK 5 Eb552 y. [ G y 5 b for some n [ Z . Examplec. In Example 8 6, we saw that Ina. particular,The set Z ofG integersis a cyclic is a group 8cyclic9 H 5groupIif there under0 is an addition. element We a [ haveG such Z6 5 that1 and G 5Z 5a .21 . b. The subgroup E 8 Z of all evenH 5integers 2 , 4is ,a 6cyclic, 8 subgroup8 Z10 of the additive group Z, generated by 2. Hence E 5 2 . H I H IH I Examplec. Inis Examplean abelian 8 6, group we saw with that respect to5 3multiplication.4 3 4 3 4 3 4 6 Since a. The set Z of integers is a cyclicH I group under addition. We have Z 5 1 and Z 5 21 . 2 3 4 b. The subgroup E 8 Z of all2 evenH55 4integers ,2 , 24 , is56 a, cyclic88, 8 2subgroupZ105 6 , of the additive group Z, generated by 2. Hence E 5 2 . H I H I c. isthenIn anExample abelian 6, group we saw with that respect3 4 3to54 3multiplication.4 33 44 3 4 33 44 6 Since3 4 3 4 H I 2 3 4 2 5H 54 , 22H, 455, 682,, 8. 2 8 5Z106 , thenis an abelian group with3 respect42 3 4to5 3multiplication.34 43 4 8333 44493 436 Since4 3 4 d. The group S(A) 5 {e, r, r , s, g, d} of Example 3 in Section 3.1 is not a cyclic group. This can be verified by considering2 a for3 all possible4 choices of a in (A). I 2 5 4 , H25 52 8., 2 5 6 , S 2 H I 83 49 d. Thethen group S(A) 5 {e, r, r3 ,4s, g,3 d4} of3 Example4 3 4 3 in3 Section4 3 4 3.1 is not a cyclic group. ExercisesThis can be verified3.3 by considering a for all possible choices of a in S(A). I H 5 2 . True or False H I 2 83 49 LabelExercisesd. The each group of the S3.3 (followingA) 5 {e, r statements, r , s, g, d as} ofeither Example true or3 infalse, Section where 3.1 His isnot a asubgroup cyclic group. of G. 1. EveryThis can group be verifiedG contains by considering at least two a subgroups.for all possible choices of a in S(A). I True or False d.2. TheThe group identity S( Aelement) = {e,σ }in in a Sectionsubgroup 3.1 H isof a acyclic group group. G must be the same as the identity Label each of the following statements asH eitherI true or false, where H is a subgroup of G. element in G. Exercises1. Every group G3.3contains at least two subgroups. 3. An element x in H has an inverse xϪ1 in H that may be different than its inverse in G. True2. The or identityFalse element in a subgroup H of a group G must be the same as the identity 4. The generator of a cyclic group is unique. Labelelement each ofin theG. following statements as either true or false, where H is a subgroup of G. 5. Any subgroup of an abelian groupϪ is1 abelian. 3.1. AnEvery element group x Gin containsH has an at inverse least two x subgroups.in H that may be different than its inverse in G. 6. If a subgroup H of a group G is abelian, then G must be abelian. 4.2. TheThe generator identity elementof a cyclic in groupa subgroup is unique. H of a group G must be the same as the identity 7.5. AnyTheelement relationsubgroup in G. R ofon an the abelian set of groupall groups is abelian. defined by HRK if and only if H is a subgroup of K is an equivalence relation. 6.3. IfAn a subgroupelement x HinofH ahas group an inverseG is abelian, xϪ1 in thenH that G must may bebe abelian.different than its inverse in G. 8. The empty set [ is a subgroup of any group G. 7.4. TheThe relation generator R onof athe cyclic set of group all groups is unique. defined by HRK if and only if H is a subgroup 9. Any group of order 3 has no nontrivial subgroups. 5. ofAnyK is subgroup an equivalence of an abelian relation. group is abelian. 10. Z under addition modulo 5 is a subgroup of the group Z under addition. 8.6. TheIf5 a emptysubgroup set [H ofis aa subgroupgroup G is of abelian, any group then G G. must be abelian. 9.7. AnyThe grouprelation of Rorderon the 3 has set no of nontrivialall groups subgroups. defined by HRK if and only if H is a subgroup 10. Zof5 underK is an addition equivalence modulo relation. 5 is a subgroup of the group Z under addition. 8. The empty set [ is a subgroup of any group G. 9. Any group of order 3 has no nontrivial subgroups. 10. Z5 under addition modulo 5 is a subgroup of the group Z under addition. Cyclic Groups - I ! Week 5 Definition 3.14 Generator 164 Chapter 3Any Groups element of a of the group G such that G = a is a generator of G . Remark. If a is a generator of G,thena21 is also, since any element x [ G can be written as x 5 an 5 (a21)2n for some integer n. Example 1 The additive group Zn 5 0 , 1 , c, n 2 1 is a cyclic group with generator [1], since5 3 4 3any4 k in3 Zn can4 6be written as k 5 k3 41 where k 1 indicates a multiple of [1] as 3described4 3 4 in Section 3.3. Elements other than [1] may also be generators. To illustrate this, consider the particular case 3 4 Z6 5 0 , 1 , 2 , 3 , 4 , 5 . The element [5] is also a generator of5 3Z46 since3 4 3 [5]4 3 is4 the3 4 additive3 4 6 inverse of [1]. The follow- ing list shows how Z6 is generated by [5]—that is, how Z6 consists of multiples of [5]. 1 5 5 5 2 5 5 5 1 5 5 4 3 354 5 354 1 5 1 5 5 3 4 354 5 324 3 4 3 4 5 354 5 314 3 4 3 4 3 4 6 354 5 304 3 4 3 4 The cyclic subgroups generated by3 4 the3 other4 elements of Z6 under addition are 0 5 0 2 5 2 , 4 , 0 833495 53346, 0 834495 5344, 324, 3046 5 2 . 83 49 53 4 3 46 Thus [1] and [5] are the only elements83 49 that53 4 are3 4 generators3 46 83 49of the entire group. I Example 2 We saw in Example 8 of Section 3.3 that H 5 2 , 4 , 6 , 8 8 Z10 forms a cyclic group with respect to multiplication53 4 3 4 3 4 3 and46 that 2 is a generator of H. The ele- ment 8 5 2 21 is also a generator of H, as the following computations confirm: 3 4 2 3 4 3 4 3 4 8 5 4 , 8 5 2 , 8 5 6 . I 3 4 3 4 3 4 3 4 3 4 3 4 Example 3 In the quaternion group G 5 {61, 6i, 6j, 6k}, described in Exercise 28 of Section 3.1, we have i2 521 i3 5 i2 ? i 52i i4 5 i3 ? i 52i2 5 1. 164 Chapter 3 Groups If a is a generator of G,thena21 is also, since any element x [ G can be written as x 5 an 5 (a21)2n for some integer n. Example 1 The additive group Zn 5 0 , 1 , c, n 2 1 is a cyclic group with generator [1], since5 3 4 3any4 k in3 Zn can4 6be written as k 5 k3 41 where k 1 indicates a multiple of [1] as 3described4 3 4 in Section 3.3.