Answer to Question #314857 in Abstract Algebra for Jyo

Question

Find the order of the following

(𝑍2 × 𝑍4)/< (1,1) >

Answer 1

$\left( 1,1 \right) +\left( 1,1 \right) =\left( 0,2 \right) \\\left( 0,2 \right) +\left( 1,1 \right) =\left( 1,3 \right) \\\left( 1,3 \right) +\left( 1,1 \right) =\left( 0,0 \right) \\from\,\,which\\\left| <\left( 1,1 \right) > \right|=\left| \left\{ \left( 1,1 \right) ,\left( 0,2 \right) ,\left( 1,3 \right) ,\left( 0,0 \right) \right\} \right|=4\\\left| \left( Z_2\times Z_4 \right) /<\left( 1,1 \right) > \right|=\frac{2\cdot 4}{4}=2$