Consider the additive group $G=Z/9Z$ x $Z/18Z$. I know that $G$ is isomorphic with $Z_9$ x $Z_{18}$ .
- Order of $(4,3)$ and $(3,5)$.
Order of $4$ in $Z_9 =9$, order of $3$ in $Z_{18}=6$ so the order of $(4,3)$ is the smallest common multiple $=18$.
Order of $(3,5)=18$.
- Is it true that $(4,3) \in \langle(3,5)\rangle$ and $(3,5) \in \langle (4,3) \rangle$?
In $Z_9, 3$ can't generate $4$ and in $Z_{18}, 3$ can't generate $5$ so none of them is true.
- Is $\{(4,3),(3,5)\}$ a system of generators for $G$?
I think it is, since 4 generates all the elements of $Z_9$ and 5 generates all the elements of $Z_{18}$.
- Is $G$ cyclic?
Since any $Z_n$ is cyclic $G$ is also cyclic and can be generated by $(5,5)$.
- Is $G/ \langle(4,3) \rangle$ cyclic ?
I don't know how to approach this…
Are these correct and are my arguments enough?
Best Answer