Find order of given factor group

Question

I'm trying to find the order of this factor group: $$(\mathbb Z_{12}\times\mathbb Z_{18}) / \langle (4,3)\rangle.$$

The order of the factor group is just the number of elements in it (aka the number of cosets). Let $H = \langle(4,3)\rangle, G = Z_{12}\times Z_{18}$.

$$H = \{(4,3), (8,6), (0,9), (4,12), (8,15), (0,0)\}$$

So a coset of $H$ in $G$ is of the form $aH$, where $a \in G$

Let $a = (0,0)$. Then $aH = H$. From above, we know this coset has 6 elements. Further, each coset must have the same number of elements.

So now we just take the order of G and divide by 6 to get the order of $G$ / $H$.

There are 12 choices for $Z_{12}$ and 18 choices for $Z_{18}$ so order of $G = 12 \cdot 18 = 216$.

$216 \div 6 = 36$. So the order of $G / H$ is 36.

Does this look right?

Thanks for the help, Mariogs

Answer 1

Yaaa. Using Lagrange’s thm Which states that if G is a finite group and H is a subgroup of G then no of left (right) cosets of H in G is |G|/|H|