Answer to Question #223551 in Abstract Algebra for Prathibha Rose

Question

Show that U(8)/is isomorphic to U(12)

Answer 1

U(8): 1, 3, 5, 7 , $3^2=5^2=7^2$ =1 mod 8

U(12): 1, 5, 7, 11, $5^2=7^2=11^2$ =1 mod 12

The map $\sigma:U(8)->U(12): \sigma(3)=5, \sigma(5)=7,\sigma(7)=11$

is $1-1,$ onto and takes the Cayley table of U(8) to that of U(12) so it is isomorphism.