In this problem we work out step-by-step various properties related to quotient groups. We will be working with congruence classes; to simply the notations we represent each congruence class by the smallest non-negative integer (so for example, we represent $3 \pmod{5}$ by the integer $3$, but not $-2$ or $8$).

Consider the finite group $U(30)$. It is an Abelian group, so every subgroup is normal. Consider the subgroup $\langle11\rangle$ of $U(30)$. Thanks to Lagrange's theorem, to compute the size of the quotient group

$$\begin{array}{cc} U(30) / \langle11\rangle \end{array}$$
it suffices to determine the size of $\langle11\rangle$. Since $\langle11\rangle$ is a cyclic group, to determine the size of $\langle11\rangle$ we need to compute the order of the generator 11, which is .

Consequently, the quotient group $U(30) / \langle11\rangle$ has size .

A complete set of coset representatives for the subgroup $\langle11\rangle$ in $U(30)$ is (make sure you enter your answer as a comma-separated list, and for each coset representative, use the SMALLEST possible positive integer)

Finally, consider the element $29$ in $U(30)$. The elements of the left coset $29\langle11\rangle$ are (make sure you enter your answer as a comma-separated list, and for each coset representative, use the SMALLEST possible positive integer)