1. ## Finding Subgroups

Let G be the group $(U(15), \times_{15})$. Find all distinct subgroup of G with exactly two elements.

Could anyone show me how to deal with this problem? So, I know that U(15)={1, 2, 4, 7, 8, 11, 13, 14} under multiplication modulo 15. I'm not sure if it helps, but I have constructed a Cayley table for G:

Any help is very appreciated.

2. Originally Posted by demode
Let G be the group $(U(15), \times_{15})$. Find all distinct subgroup of G with exactly two elements.

Could anyone show me how to deal with this problem? So, I know that U(15)={1, 2, 4, 7, 8, 11, 13, 14} under multiplication modulo 15. I'm not sure if it helps, but I have constructed a Cayley table for G:

Any help is very appreciated.

Of course it helps! A subgroup with two elements is of the form $\{1,a\}\,,\,\,with\,\,\,a^2=1$ , so locate all the elements with order 2 and

Tonio

3. And, of course, to do that you just have to look down the diagonal in your table.

4. Okay then it's all the 1's in the main diagonal: g={1, 4, 11, 14}.

And elements like 2 belong to order 4? Because $2 \times_{15} 2 = 4, 4 \times_{15} 2 = 8, 8 \times_{15} 2 = 1$. Is that right?

5. Originally Posted by demode
Okay then it's all the 1's in the main diagonal: g={1, 4, 11, 14}.

And elements like 2 belong to order 4? Because $2 \times_{15} 2 = 4, 4 \times_{15} 2 = 8, 8 \times_{15} 2 = 1$. Is that right?

Right indeed.

Tonio