Question

How many generators of the cyclic group of order 8 are
(a) 1
(b)2
(c) 4
(d) 5​

Answer 1

Answer:

5

Step-by-step explanation:

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Answer 2

The number of generators of the cyclic group of order 8 are 4

Given :

A cyclic group of order 8

To find :

The number of generators of the cyclic group of order 8 are

(a) 1

(b) 2

(c) 4

(d) 5

Solution :

Step 1 of 2 :

Write down the given group

Let G be the cyclic group of order 8

Next suppose that let G is generated by a such that G = < a >

Then o(a) = | G | = 8

Step 2 of 2 :

Find number of generators of the cyclic group

Let aⁿ is also a generator of G

We know that if aⁿ is a generator of G of order m if gcd(m, n) = 1

Since aⁿ is also a generator of G of order 8

Then gcd(8, n) = 1

⇒ n = 1 , 3 , 5 , 7

∴ a , a³ , a⁵ , a⁷ are generators of G of order 8

Thus number of generators of the cyclic group of order 8 are 4

Hence the correct option is (c) 4

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