Originally Posted by

**Deveno** U(11) (the group of units mod 11, which consists of all the non-zero elements of Z11, under multiplication mod 11), is a group of order 10. as it turns out, this group is also cyclic, although that is not obvious.

let's see if 2 is a generator:

2^{2} = 4

2^{3} = 8

2^{4} = 16 = 5 (mod 11)

2^{5} = (5)(2) = 10 (mod 11)

2^{6} = (10)(2) = 20 = 9 (mod 11)

2^{7} = (9)(2) = 18 = 7 (mod 11)

2^{8} = (7)(2) = 14 = 3 (mod 11)

2^{9} = (3)(2) = 6 (mod 11)

2^{10} = (6)(2) = 12 = 1 (mod 11), and 1 is the identity of U(11), so we see the order of 2 is 10, so 2 is indeed a generator.

now...why does this tell us that 4 and 10 are NOT generators?

you go ahead and check 3, and then 5.