What is the generator of the cyclic group (Z,+)?

In summary, the conversation discusses the concept of a cyclic group, specifically in the context of the group of integers under addition. The cyclic subgroup generated by an element is defined as the set of all powers of that element, including negative powers in the case of an additive group. Therefore, both 1 and -1 are generators of the group of integers. It is also mentioned that in an infinite cyclic group, an element and its inverse have the same order and are the only possible generators.
  • #1
LagrangeEuler
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I do not understand why ##(Z.+)## is the cyclic group? What is a generator of ##(Z,+)##?
If I take ##<1>## I will get all positive integers. If I take ##<-1>## I will get all negative integers. I should have one element which generates the whole group. What element is this?
 
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  • #2
LagrangeEuler said:
I do not understand why ##(Z.+)## is the cyclic group? What is a generator of ##(Z,+)##?
If I take ##<1>## I will get all positive integers. If I take ##<-1>## I will get all negative integers. I should have one element which generates the whole group. What element is this?
The positive integers do not form a subgroup, so they cannot be the subgroup generated by the element ##1##. Something is wrong with the definitions you are using.
 
  • #3
If ##(G, \cdot)## is a group and ##g\in G##, the cyclic subgroup generated by ##g## is ##\langle g\rangle = \{g^n : n\in \mathbb{Z}\}##, where ##g^0 = e##, ##g^n= g\cdot g\cdots g## (##n## times) if ##n## is a positive integer and ##g^n = g^{-1}\cdot g^{-1}\cdots g^{-1}## (##-n## times) is ##n## is a negative integer.

If ##G## is additive, ##ng## is written in place of ##g^n##. So when ##G## is additive ##\langle g\rangle = \{ng : n\in \mathbb{Z}\}##. In the case ##G = \mathbb{Z}##, the cyclic subgroup ##\langle 1\rangle = \{n\cdot 1 : n\in \mathbb{Z}\} = \{n : n\in \mathbb{Z}\} = \mathbb{Z}##. Similarly ##\langle -1\rangle = \mathbb{Z}##. So indeed, ##\mathbb{Z}## is cyclic, and both ##-1## and ##1## are generators of the group.
 
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  • #4
Euge said:
If ##G## is abelian,
I think you mean additive.
 
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  • #5
PeroK said:
I think you mean additive.
Corrected, thanks!
 
  • #6
Can I add, as a non-mathematician….

If a finite group is cyclic, the group can be generated by a single element.

But if an infinite group (such as integers under addition) is cyclic, the group can be generated by a single element or its inverse. So in the present case we can use 1 or -1.
 
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  • #7
Steve4Physics said:
Can I add, as a non-mathematician….

If a finite group is cyclic, the group can be generated by a single element.

But if an infinite group (such as integers under addition) is cyclic, the group can be generated by a single element or its inverse. So in the present case we can use 1 or -1.
A group element has the same order as its inverse, so for any group ##G## and ##g\in G##, the cyclic subgroup ##\langle g\rangle = \langle g^{-1}\rangle##.
 
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  • #8
A fun exercise is proving in an infinite cyclic group that ##g## and ##-g## are the only generators, which is not true for finite groups - e.g. the cyclic group of order ##p## for ##p## prime has every element as a generator except for the identity.
 
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Related to What is the generator of the cyclic group (Z,+)?

What is a cyclic group?

A cyclic group is a type of mathematical group that is generated by a single element, called a generator. This means that all other elements in the group can be obtained by repeatedly applying the group operation to the generator.

Why is (Z,+) a cyclic group?

(Z,+) is a cyclic group because it is generated by a single element, 1. This means that all other integers can be obtained by repeatedly adding 1 to itself, which is the group operation in (Z,+).

How do you prove that (Z,+) is a cyclic group?

To prove that (Z,+) is a cyclic group, you can show that every element in the group can be written as a power of the generator, 1. This can be done by using mathematical induction and showing that the generator can be used to obtain all other elements in the group.

What are some applications of cyclic groups?

Cyclic groups have many applications in mathematics, computer science, and cryptography. They are used in coding theory, error-correcting codes, and cryptographic algorithms such as the Diffie-Hellman key exchange and the RSA encryption scheme.

Can any other group be cyclic?

No, not all groups can be cyclic. In order for a group to be cyclic, it must have a single element that can generate all other elements. This is not true for all groups, such as non-abelian groups and infinite groups.

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