isomorphisms

[Benzoate]

1. The problem statement, all variables and given/known data

Prove or disprove U(20) and U(24) are isomorphic

2. Relevant equations



3. The attempt at a solution

U(20)={1,3,,7,9,11,13,17,19}
U(24)={1,5,7,11,13,17,19}

In U(20), Order(1)=1 Order(3)=4 Order(7)=4 Order(9)=2, Order(11)=2 Order(13)=4, Order(17)=4, Order(19)=2

In U(24), Order(1)= 1 , Order(5)=2 , ORder(7)=2, Order(11)=2 , Order(13)=2 , Order(17)=2, Order(19)= 2

Since each of the elements in each of the groups do not generate the same order for each elements in the opposite group, can't I conclude thatt U(24) and U(20) are not isomorphic?

[morphism]

You can say that U(24) has 6 elements of order 2, while U(20) doesn't. So they certainly can't be isomorphic.

(By the way, you're missing 23 in U(24). So in fact U(24) has 7 elements of order 2, but this is immaterial.)