Proving Z is Abelian Group Under Normal Addition

In summary, the conversation discusses whether Z, the set of all integers, is an abelian group under normal addition. It is mentioned that the definition of Z does not explicitly state that it is an abelian group, but rather it is typically proved to be one. This is shown through the distributive law and the commutative property of addition. The distributive law is an axiom in the definition of a ring, but it can also be proved to hold for the natural numbers, integers, and rational numbers.
  • #1
pivoxa15
2,255
1
Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?
 
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  • #2
pivoxa15 said:
Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?

What exactly is your question? (Z, +) is an abelian group. I don't believe the integers are defined that way. They only "fit into this given definition".
 
  • #3
pivoxa15 said:
Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?
First, the "natural numbers" are NOT Z. The natural numbers include only the positive integers (some texts include 0) while Z is all integers. Obviously the natural numbers does NOT form a group since they do not have additive inverses. Typically, Z is defined in terms of natural numbers (say, as equivalence classes of pairs of natural numbers) and then the fact that they form an abelian group is proved.
 
  • #4
That + is abelian follows from

(x + y)z = xz + yz
x(y + z) = xy + xz (both proved by Peano i think)

Given a, b in Z.

(a + b)(1 + 1) = a(1 + 1) + b(1 + 1) = a + a + b + b,
(a + b)(1 + 1) = (a + b)1 + (a + b)1 = a + b + a + bthis implicates

a + b + a + b = a + a + b + b

so

a + b = b + a

so + is abelian
 
Last edited:
  • #5
Ultraworld said:
That + is abelian follows from

(x + y)z = xz + yz
x(y + z) = xy + xz (both proved by Peano i think)

Given a, b in Z.

(a + b)(1 + 1) = a(1 + 1) + b(1 + 1) = a + a + b + b,
(a + b)(1 + 1) = (a + b)1 + (a + b)1 = a + b + a + b


this implicates

a + b + a + b = a + a + b + b

so

a + b = b + a

so + is abelian

This shows that it's not necessary to assume that, in a ring with addition, addition is commutative, since it follows from the other ring axioms.

The distributive law is an axiom, btw, there's nothing to prove.
 
  • #6
An axiom in what system? Certainly the distributive law is part of the definition of "ring" and so a axiom in that sense. The fact that the distributive law is true for the natural numbers, integers, rational numbers, etc. can be proved.
 
  • #7
HallsofIvy said:
An axiom in what system? Certainly the distributive law is part of the definition of "ring" and so a axiom in that sense.

Yes, that's what I meant. I assumed Ultraworld was referring to a ring, since there's multiplication and addition in his post.
 

1. What is an Abelian group?

An Abelian group is a mathematical structure that satisfies the commutative property, meaning that the order of operations does not affect the outcome. In other words, when two elements of the group are combined using a binary operation, the result is the same regardless of the order in which the elements are combined.

2. What does it mean for a group to be "under normal addition"?

For a group to be "under normal addition" means that the group operation being used is addition, and that the elements of the group are being operated on using the usual rules of addition (such as commutativity, associativity, and the existence of an identity element).

3. How do you prove that Z (the integers) is an Abelian group under normal addition?

To prove that Z is an Abelian group under normal addition, you would need to show that the group satisfies the four axioms of a group: closure, associativity, identity element, and inverse element. You would also need to demonstrate that the group operation (addition) is commutative, meaning that the order of operations does not affect the outcome.

4. What are some examples of non-Abelian groups under normal addition?

Some examples of non-Abelian groups under normal addition include matrices, quaternions, and octonions. In these structures, the order of operations does affect the outcome, meaning that they do not satisfy the commutative property and are therefore not Abelian groups.

5. Why is proving Z is an Abelian group under normal addition important?

Proving that Z is an Abelian group under normal addition is important because it is a fundamental concept in mathematics. It helps to establish a foundation for understanding more complex mathematical structures and can be applied in various fields such as algebra, number theory, and cryptography. Additionally, understanding the properties of Abelian groups can help to develop problem-solving skills and logical reasoning abilities.

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