Groups and Subgroups elements

Question

Let $G$ be a cyclic group order $12$ with $G=\left<a\right>$. Let $H=\left<a^3\right>$. List the elements of $H$ and find the cosets. I am lost as to what the elements of $H$ would be. Is it just $a^3, a^6, a^9$?

Answer 1

$H = \{1, a^3, a^6, a^9\}$. There are $[G:H]=|G|/|H|=12/4=3$ cosets.

The cosets are:

$H=\{1, a^3, a^6, a^9\}$

$aH=\{a, a^4, a^7, a^{10}\}$

$a^2H=\{a^2, a^5, a^8, a^{11}\}$

So, $G/H = \{H, aH, a^2H\}$

We see that:

$H=a^3H=a^6H=a^9H$

$aH=a^4H=a^7H=a^{10}H$

$a^2H=a^5H=a^8H=a^{11}H$