Understanding the Abstract: Defining Groups in Abstract Algebra

[Grouphelp]

So I'm in Abstract Alegebra I, straight out of Foundations of Math, what most people call discrete math, so I can manipulate sets of numbers very easily.

My basic question: What really is a group? I've been in this class for a while and everyone talks very fluently in group theory, I understand the requirements to be a group, but is it a bunch of sets? I need a better definition than just the three requirements, can someone help?

 

[Landau]

A bunch of sets? Where do you get that from? A group is itself a set X, together with structure: an associative binary operation X x X -> X, an identity element e\in X, and each element of X has an inverse.

I don't know what you mean by a "better" definition. Perhaps "a category with one object, and all of whose arrows are invertible"? Or "a groupoid with one object"? Or "a monoid where every element has an inverse"? These are of course all equivalent; rephrasings of the same definition.

 

[Fredrik]

I don't understand what you're asking for. The definition is simple: A pair $(G,\star)$ is said to be a group if $\star$ is a binary operation on G and

(a) $(x\star y)\star z=x\star(y\star z)$ for all x,y,z in G.
(b) There's an element e in G such that $x\star e=e\star x=x$ for all x in G.
(c) For each x in G, there's an element $x^{-1}$ in G such that $x^{-1}\star x=x\star x^{-1}=e$.

Are you looking for examples of groups? There are usually lots of examples in math books that cover this sort of stuff.

 

[Grouphelp]

Sheesh. Such mouth. I understand you all get this stuff but I'm slightly confused, forgive me. I can tell you two are not teachers. I'm simply asking a clarification, let me be clearer:

I'm a student who can manipulate groups in problems and knows the steps, I'm just having a mental lapse about understanding what a group really is and let's face it, my textbook is next to worthless BECAUSE this topic is so easy, and I don't quite get why I'm struggling with this either. Thank you for rewording exact definitions, but what I'm asking is more like this:

What makes the set of all integers different from the group (Z, +)?
I understand the set of all integers to be the set containing the elements of all the positive integers, i.e., (...-3, -2, -1, 0, 1, 2, 3, ...), but what are the "elements" of (Z, +)? To me, based on the definition, the set of Z contains an identity element (0) all of the inverses are already present and it is associative. But what does the "+" do and how is (Z, +) different from Z?

 

[Fredrik]

What makes the set of all integers different from the group (Z, +)?

You may not like this answer either, but I don't know what else I can say except "addition". If Z is the set of integers, the pair (Z,+) is a group, so really, the only difference is that you have paired the set with the addition operation.

I understand the set of all integers to be the set containing the elements of all the positive integers, i.e., (...-3, -2, -1, 0, 1, 2, 3, ...), but what are the "elements" of (Z, +)?

Ah, that's something that requires clarification. Z is said to be the "domain", the "universe", or the "underlying set" of the group (Z,+). When people talk about the members of a group, they are really referring to the members of the underlying set. So a member of (Z,+) is by definition a member of Z. A related issue is that everyone defines a group as a pair (G,*) with certain properties, but as soon as they're done with the definition, they start referring to the underlying set G as a "group", even though that's technically incorrect. I think almost everyone finds this acceptable since it rarely causes any confusion once you have understood that this is what people are doing.

There's also another answer, which doesn't have anything to do with groups. It's the issue of how to define ordered pairs using a theory of unordered sets. I'm not going to elaborate on that unless you ask, because I'm assuming that that's not what bothers you here.

 

[lavinia]

Sheesh. Such mouth. I understand you all get this stuff but I'm slightly confused, forgive me. I can tell you two are not teachers. I'm simply asking a clarification, let me be clearer:

I'm a student who can manipulate groups in problems and knows the steps, I'm just having a mental lapse about understanding what a group really is and let's face it, my textbook is next to worthless BECAUSE this topic is so easy, and I don't quite get why I'm struggling with this either. Thank you for rewording exact definitions, but what I'm asking is more like this:

What makes the set of all integers different from the group (Z, +)?
I understand the set of all integers to be the set containing the elements of all the positive integers, i.e., (...-3, -2, -1, 0, 1, 2, 3, ...), but what are the "elements" of (Z, +)? To me, based on the definition, the set of Z contains an identity element (0) all of the inverses are already present and it is associative. But what does the "+" do and how is (Z, +) different from Z?

Z is just the set of all integers together with the operation of addition. There is no difference between the set of integers and (Z,+). By itself as only a set, Z is just a bunch of names. To make it into a group you need addition. Addition does not make Z into another set and (Z,+) is not another set. It is the original set together with the operation of addition. Addition is actually a mapping from pairs of integers back to the integers e.g. (5,3) -> 8.

If you wanted to, but nobody ever does, you could think of this addition mapping as a set. It would be the set of pairs, ((n,m), n+m).

 

[Wizlem]

I'm in Abstract Alegebra

there's a reason it's called ABSTRACT. The biggest problem you seem to be having is cutting away the real world thinking from the topic. If you can manipulate groups just find then that would suggest you understand the material. Overthinking it is what seems like the problem to me. This kind of thing always reminds me of a quote. "One must be able to say at all times -- instead of points, lines, and planes -- tables, chairs, and beer mugs." -- David Hilbert