Question

Prove that G =({0, 1, 2, 3, 4}, +5 ) is a cyclic group with generators 1 and 4.​

Answer 1

Answer:

19 is the your best answer

Answer 2

Concept:

Cyclic group: In group theory, a cyclic group is a group that is generated by a single element.

Given:

We are given that:  G =({0, 1, 2, 3, 4}, +5 )

Find:

We need to prove that G =({0, 1, 2, 3, 4}, +5 ) is a cyclic group with generators 1 and 4.​

Solution:

First we need to prove that G =({0, 1, 2, 3, 4}, +5 ) is a cyclic group:

For this, we can see that 0,1,2,3,4 are 5 elements.

n(0,1,2,3,4)≤5 is true.

Also {0,1,2,3,4}, all are co-prime to 5.

Since, both the properties hold true, it proves that the group is a cyclic group.

Now, we need to prove that it has generators 1 and 4:

Since the HCF of 1 and 5 is 1 and the HCF of 4 and 5 is also 1, it proves that both 1 and 4 are the generators of the cyclic group.

Therefore, it is being proved that G =({0, 1, 2, 3, 4}, +5 ) is a cyclic group with generators 1 and 4.​

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