For n 20 which groups Un are cyclic Make a conjecture as to

For n 20, which groups U(n) are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?

Solution

The groups U(n) are the groups with respect to multiplication modulo n.

U(2)={1} is cyclic as 1 is generator of the group.

U(3)={1,2} is also cyclic as 2 is generator of the group. Similarly,U(4) is cyclic with 3 as generator. U(5) is also cyclic with 2 and 3 as generators. And similarly U(10),U(14) are cyclic but U(8),U(12),U(15),U(16),U(20) are not cyclic.

One have to solve these to know wheather a group is cyclic or not but there are some theorems like U(2^n) (n>2) is not cyclic.

PROOF- Let us take two elements (2^n)-1 and (2^n-1)+1 two elements of U(2^n).(You can check in any U(2^n) (n>2) group these two elements exists)

The element (2^n)-1 is of order 2.Therefore,

((2^n)-1)^2=1 mod (2^n) and

((2^n-1)+1)^2=1+2^(2n-2)+2^(n)=1 mod 2^(n)

so,distint element have same order which is not possible in case of cyclic group.

Therefore,U(2^n) (n>2) is not cyclic.