- #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?
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.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?
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
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.
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.
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).
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.
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.
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.