I'm trying to find the order of an element in the quotient group: $\mathbb{Z_{20}} \times U(15) / \langle (15, 7)\rangle$. Call $G$ the the external direct product and $H = \langle(15, 7)\rangle$
So what I did was find $|G| = 160$ and $|H| = 4$ so $|G/H| = 40$.
So $\implies |(16,1)H| \in \{1, 2, 4, 5, 8, 10, 20, 40\}$.
Then I listed out $H$ : $\{(15, 7), (10, 4), (5, 13), (0, 1)\}$.
Now $(16,1)H \neq H$ since $(16,1) \notin H$.
But $5(16,1) = (80, 1) = (0, 1)$
So since $(0, 1) \in H, 5[(16,1)H] = (0,1)H = H$.
$\therefore |(16,1)H| = 5$.
Can someone confirm?