# Cyclic Groups and Generators

**Posted:**June 9, 2020

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Some groups have an interesting property: all the elements in the group can be obtained by repeatedly applying the group operation to a particular group element. If a group has such a property, it is called a cyclic group and the particular group element is called a generator. A trivial example is the group *Z*_{n}, the additive group of integers modulo *n*. In *Z*_{n}, 1 is always a generator:

1 ≡ 1 mod n

1+1 ≡ 2 mod n

1+1+1 ≡ 3 mod n

…

1+1+1+…+1 ≡ n ≡ 0 mod n

If a group is cyclic, then there may exist multiple generators. For example, we know *Z*_{5} is a cyclic group. The element 1 is a generator for sure. And if we take a look at 2, we can find:

2 ≡ 2 mod 5

2+2 ≡ 4 mod 5

2+2+2 ≡ 6 ≡ 1 mod 5

2+2+2+2 ≡ 8 ≡ 3 mod 5

2+2+2+2+2 ≡ 10 ≡ 0 mod 5

So all the group elements {0,1,2,3,4} in *Z*_{5} can also be generated by 2. That is to say, 2 is also a generator for the group *Z*_{5}.

Not every element in a group is a generator. For example, the identity element in a group will never be a generator. No matter how many times you apply the group operator to the identity element, the only element you can yield is the identity element itself. For example, in *Z*_{n}, 0 is the identity element and 0+0+…+0 ≡ 0 mod n in all cases.

Not every group is cyclic. For example,* Z _{n}^{*}*, the multiplicative group modulo n, is cyclic if and only if

*n*is 1 or 2 or 4 or

*p*or

^{k}*2*p*for an odd prime number

^{k}*p*and

*k*≥ 1. So

*Z*must be a cyclic group because 5 is a prime number. Actually all the elements in

_{5}^{*}*Z*

_{5}

^{*}, {1,2,3,4} can be generated by 2:

2^{1} ≡ 2 mod 5

2^{2} ≡ 4 mod 5

2^{3} ≡ 8 ≡ 3 mod 5

2^{4} ≡ 16 ≡ 1 mod 5

And *Z _{12}^{*}* is not a cyclic group. The elements in

*Z*are: {1,5,7,11}. Obviously the identity element 1 cannot be a generator. Let’s check the other three elements:

_{12}^{*}5^{1} ≡ 5 mod 12 |
7^{1} ≡ 7 mod 12 |
11^{1} ≡ 11 mod 12 |

5^{2} ≡25 ≡ 1 mod 12 |
7^{2} ≡ 49 ≡ 1 mod 12 |
11^{2} ≡ 121 ≡ 1 mod 12 |

None of the elements can generate the whole group. Therefore, none of them is a generator. So *Z _{12}^{*}* is indeed not cyclic.

If *Z _{n}^{*}* is cyclic and

*g*is a generator of

*Z*, then

_{n}^{*}*g*is also called a primitive root modulo

*n*.

Source: https://www.doc.ic.ac.uk/~mrh/330tutor/ch06.html

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