4 votes 4 votes Let $Z$ be the set of all integers. Let $n \in Z$ and $nZ = \{nk : k \in Z \}.$ We know that $Z$ is a group under addition operation. Which of the following is/are true? $nZ$ is a subgroup of $Z$ (under addition operation) for all $n \in Z.$ Every subgroup of $(Z,+)$ is isomorphic to $nZ$ for some $n.$ $3Z$ is the smallest subgroup of $(Z,+)$ containing $3.$ $nZ$ is a cyclic subgroup of $Z.$ Set Theory & Algebra goclasses_wq12 goclasses set-theory&algebra group-theory group-isomorphism multiple-selects 2-marks + – GO Classes asked May 29, 2022 edited May 3, 2023 by Deepak Poonia GO Classes 509 views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply rhl commented Jun 5, 2022 reply Follow Share Can anyone explain how option B is true? what will happen in the case if we take the subgroup ( 0 , + )? 0 votes 0 votes shishir__roy commented Jun 5, 2022 reply Follow Share it will be isomorphic to nZ such that n = 0. 1 votes 1 votes Please log in or register to add a comment.
3 votes 3 votes All the statements are true. We have all these in our lectures. $(Z,+)$ is a cyclic group, and every subgroup of a cyclic group is cyclic. GO Classes answered May 29, 2022 GO Classes comment Share Follow See all 2 Comments See all 2 2 Comments reply abinah13aeccse commented May 31, 2022 i reshown by abinah13aeccse May 31, 2022 reply Follow Share I have a doubt in option C , How are we comparing size of infinte subgroups – how 3Z is smallest..?? what is the condition of saying it smallest subgroup.? 0 votes 0 votes shishir__roy commented May 31, 2022 reply Follow Share @abinah13aeccse refer to the definition of subgroup generated by an element. 0 votes 0 votes Please log in or register to add a comment.