2.2. SUBGROUPS AND CYCLIC GROUPS 95 denoted h S i . When S D f a g is a singleton, the subgroup generated by S is denoted by h a i . We say that G is generated by S or that S generates G if G D h S i . A “constructive” view of h S i is that it consists of all possible prod-ucts g 1 g 2 ±±± g n , where g i 2 S or g ± 1 i 2 S . Another view of h S i , which is sometimes useful, is that it is the intersection of the family of all sub-groups of G that contain S ; this family is nonempty since G itself is such a subgroup. The family of subgroups of a group G are partially ordered by set inclusion. 1 In fact, the family of subgroups forms what is called a lattice . 2 This means that, given two subgroups A and B of G there is a unique smallest subgroup C such that C ² A and C ² B . In fact, the subgroup C is h A [ B i . Furthermore, there is a unique largest subgroup D such that D ³ A and D ³ B ; in fact, D D A \ B . Cyclic Groups and Cyclic Subgroups
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.