Find all the cosets of $A=[a^3]$ in a cyclic group $B=[a]$ if we know that order of $B$ is $12.$

Question

I was trying to solve the following question:

Find all the cosets of $A=[a^3]$ in a cyclic group $B=[a]$ if we know that order of $B$ is $12.$

But I am confused about $a^3$. Could you point me in the right direction for the solution?

Similarly, there is a subproblem:

Find all the cosets of $H=[3]$ in the modular additive group $\Bbb Z_{12}$

Answer 1

The cosets are $[a^3], a[a^3], a^2[a^3], a^3[a^3]$ since $a^4[a^3]=\{a^4b\mid b\in [a^3]\}=[a^3]$ as $a^{12}=e$ (and the quotient of a cyclic group is cyclic).

For the subproblem, establish the isomorphism $\varphi$ given by $a\mapsto [1]_{12}$.