Why are only non-commutative groups called non-abelean or is this wrong?

[bentley4]

According to wiki:
"a non-abelian group, also sometimes called a non-commutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a."
I thought in order to be an abelean group, 5 axioms must be satisfied. If one of them is not satisfied it would logically be a non-abelean group. Then why is it only called non-commutative group? Is this just a bad name or do I misunderstand?

 

[micromass]

I don't quite understand your question. Noncommutative groups are exactly thesame thing as non-abelian groups.
A non-abelian group must satisfy the group axioms and must have at least two elements that do not commute. So there are 5 axioms to satisfy...

 

[bentley4]

Thnx, that is what I wanted to know.

 

[HallsofIvy]

More correctly "commutative" (named for the fact that you can commute the elements in the group operation) and "Abelian" (named for Henrik Abel) mean exactly the same thing. Therefore, "non-commutative" and "non-Abelian" mean exactly the same thing.

 

[bentley4]

Owk. So you just reasoned that non-commutative and non-Abelian are the same because of the concepts commutative and non-commutative are jointly exhaustive and mutually exclusive(as abelian and non-abelian). This would've only be usefull if I assumed abelian and commutative were the same, which I didn't. But I still appreciate your input though. The problem is that I thought a commutative group was a group with only the commutative property and nothing else. But I was wrong apparently since it also has closure, associativity , an identity element and an inverse element.

 

[Stephen Tashi]

I thought a commutative group was a group with only the commutative property and nothing else. But I was wrong apparently since it also has closure, associativity , an identity element and an inverse element.

The problem is how you interpret the statement that something "was a group with only the commutative property", but it's an understandable problem. The normal usage of English allows us to assign meanings to individual words, so a "commutative group" is a thing that was both commutative and also a group. Since it's a group, it must have closure, associativity, an identity and inverses.

However, one cannot always assume that definition of a mathematical object that is named by several words can be analyzed by defininig each word individually. For example: "The limit of the function f(x) as x approaches A is L" had a definition. But you can't retrieve the complete definition by analyzing the meaning of the individual words. To me the clearest type of mathematical statement is one that takes the form:
Statement R means statement S. So the definition of a group could be written in the form: "G is a group means that G is a set of elements with a binary operation ... " etc.
This makes it clear that you aren't supposed to analyze the meaning of the phrase that is defined by parsing the individual words in it. But most people find writing mathematical definitions in that form too awkward sounding or they estimate that their readers will find that style difficult to understand.

 

[bentley4]

Again a splendid post Stephen! Thank you

However, one cannot always assume that definition of a mathematical object that is named by several words can be analyzed by defininig each word individually.

This arbitrariness is very confusing if you are new to the math world. Because you have to start somewhere, math in words. The more you advance in math the more you can substitute words for symbols and the less ambiguous things get. So once you have understood a certain basis, this ambiguity (almost?) completely resolves. Thats why a lot of professional mathmaticians forget this problem I think. I understand it would be too hard for a change of certain terms to be adopted so there would be at least a uniform interpretation such as that in titles every part of a title would be true for the whole. The alternative is that mathematicians should make a sort of introduction that point out exactly where these ambiguous interpretations lie at least for the mathmatical foundations, set theory. Knowing that you realize this ambiguity so clearly soothes me.

To me the clearest type of mathematical statement is one that takes the form:
Statement R means statement S. So the definition of a group could be written in the form: "G is a group means that G is a set of elements with a binary operation ... " etc.
But most people find writing mathematical definitions in that form too awkward sounding or they estimate that their readers will find that style difficult to understand.

I would be curious to read such a text. It might be boring to read but this should be used as a reference in the case of confusion. This would be very helpful for beginning math students I think.