The relation which we see between the orders of the elements of a group and the order of the group is smallest positive power.
G = U(20), order of each element of G = ?
The smallest positive power of an element that gives you your identity is the order of the element in a group. Three instances are covered: complex numbers, a 2x2 rotation matrix, and real number elements of finite order.
G = U(20) = {1,3,7,9,11,13,17,19}
⇒ 3° ≡ 1, 3¹ ≡ 3, 3² ≡ 9, 3³ ≡ 7, 3¹ ≡ 1(20)
∴ order of 3 = 4
7⁴ ≡ 1 (mod 20) → order (7) = 4
9² ≡ 1 (mod 20) → order (9) = 2
11² ≡ 1 (mod 20) → order (11) = 2
13² ≡ 9(mod 20) → 13⁴ ≡ 81(mod 20) → 13⁴ ≡ 1 (mod 20)
⇒ order (13) = 4
order 07 = 4
17 ≡-3(mod 20) ⇒ 17² ≡ yy (mod 20)
⇒ 17⁴ ≡ 81 (mod 20) ≡ 1 (mod 20) ⇒
19 = -1 (mod 20)
19² ≡ 1 (mod 20) ⇒ order (19) = 2
Hence we get the requires answer.
Learn more about orders here:
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