4 votes 4 votes The group $Z_{n}$ consists of the elements $\{0,1,2, \ldots, n-1\}$ with addition $\bmod n$ as the operation.How many subgroups of $Z_{9}$ are there? Set Theory & Algebra goclasses_wq12 numerical-answers goclasses set-theory&algebra group-theory 2-marks + – GO Classes asked May 29, 2022 GO Classes 336 views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
3 votes 3 votes NOTE that $Z_{n}$ is a cyclic group, hence an abelian group. For any group, we know by Lagrange's theorem that the order of subgroup divides the order of the group. For abelian groups, the converse of Lagrange's theorem is also true. For the Abelian group, we know that if $d$ divides the order of the group then there exists a subgroup of size $d.$ The subgroups of $Z_9$ are: $\{0\},$ $\{0, 3, 6\},$ $Z_{9}.$ GO Classes answered May 29, 2022 • edited May 29, 2022 by Lakshman Bhaiya GO Classes comment Share Follow See all 0 reply Please log in or register to add a comment.
