Question

The number of generator of a finite cyclic group of order 50 is
1. 8
2. 13
3. 17
4. 20

Answer 1

Correct Answer - Option 4 : 20

Concept:

If a cyclic group G is generated by an element 'a' of order 'n', then am is a generator of G if m and n are relatively prime.

Calculation:

Let a cyclic group G of order 50 generated by an element a, then

o(a) = o(G) = 50;

to determine the number of generator of G evidently,

G = {a, a2, a3, ........ , a⁵⁰ = e}

An element am € G is also a generator of G is H.C.F of m and 50 is 1.

H.C.F of (1, 50) is 1 silimarly, (3, 50) (7, 50), (9, 50) (11, 50), (13, 50), (17, 50), (19, 50), (21, 50), (23, 50), (27, 50), (29, 50), (31, 50), (33, 50), (37, 50), (39, 50), (41, 50), (43, 50), (47, 50), (49, 50)

Therefore, there are 20 generators of G.