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Permutation group operation

I find the example of "A finite nonabelian group: permutations of a set' to be confusing and wonder if someone could edit it to make it clearer. In particular, what is the operation of this group? What are the elements of the group? Previous examples had a clear operation (addition of integers, etc). It's not clear what the operation is here. Is it a and b? If so, doesn't that mean this group has 2 operations not just 1. Or is it, "then do". as in ab means b then do a. If the permutations are the elements of the group, then what are a and b?

Art Salwin (Talk) 21:17, 14 July 2005

A right identity and right inverses suffice

How do you prove the left inverse and left identity axioms from the right inverse and right identity axioms?

 $a*a^{-1}=e$, 
 $a*a^{-1}*a=e*a$,
 $e*a*(a^{-1}*a)^{-1}=a*a^{-1}*a*(a^{-1}*a)^{-1}=a*e=a$,
 $e=a^{-1}*a*(a^{-1}*a)^{-1}=a^{-1}*e*a*(a^{-1}*a)^{-1}=a^{-1}*a$

 $a*e=a$, $a*a^{-1}*a=a$, $e*a=a$

There seems to be a mistake in this reasoning. This shows that $a*a^{-1}$ behaves like a right-identity for $a$. However, this is not necessarily the same element for all $a$. In fact, there is a counter-example to this proposition. Consider the two-element structure {1, 2} whose binary operation satisfies the rule $x*y=x$. Both elements of this structure are a right identity, and thus this structure satisfies the right identity and right inverse axioms. It also satisfies associativity, but it does not have a left identity.

Your counterexample is wrong: whichever element is your chosen right identity element, the other element will have no right inverse with respect to that right identity. --Zundark 18:12, 3 August 2006 (UTC)

Whether the (not mine) counterexample is right depends on the exact form of the right identity and right inverse axioms. The current formulation "right identity (that is, there is an element e such that x * e = x for all x) and right inverses with respect to this right identity (that is, for each x there is a y such that x * y = e)" does not make sense as the statement "there is an element e such that x * e = x for all x" that is $\exists _{e\in G}\forall _{x\in G}x\cdot e=x$ is equivalent to "$\{e\in G:\forall _{x\in G}\ x\cdot e=x\}\neq \emptyset$". Then what does "this" refers to in "right inverses with respect to this right identity"? Using "the" in the standard group definition (the neutral element from the previous axiom) was (sort of) acceptable, because one can prove from the second axiom that there is exactly one neutral element. As shown by the example above, this is not the case for associativity + existence of a right neutral element. --Slawekk 18:57, 3 August 2006 (UTC)

Let S be a semigroup with a right identity e. If every element of S has a right inverse with respect to e, then S is a group. If you want to reword the article to make it clearer, then do so. --Zundark 19:36, 3 August 2006 (UTC)

I added "unique" to the "...right identity (that is, there is an element e such that...". This is weaker that what Zundark says, but true and illustrates the point that one can weaken the standard axioms. --Slawekk 21:42, 3 August 2006 (UTC)

Let me address the larger issue: the axioms of right identity and right inverses are not actually independent of each other, even in the traditional two-sided formulation. The right identity axiom does not merely assert that the set of right identities is nonempty, it asserts that one of them is distinguished. The axiom of right inverses refers to the distinguished one mentioned in the previous axiom.

One way of getting around the difficulties Slawekk raises is to realize that there is really one axiom here, and it goes as follows: There exists $e\in G$ such that ((for all $x\in G,x*e=x$) AND (for each $x\in G$, there exists $x^{-1}\in G$ such that $x*x^{-1}=e$)).

A different solution to the problem is the universal algebra approach. Instead of thinking of a group as a set with a binary operation (G,*), it is really a set with a binary operation, a unary operation, and a distinguished element (G,*,-1,e). This works for either the traditional two-sided formulation or the one-sided formulation to which the paragraph we're discussing alludes. There are technical advantages to this approach. For instance, one doesn't have to say "there exists e" in the right identity axiom because the distinguished element is already part of the data; instead, one just writes x*e = x for all x. In the traditional two-sided formulation, the uniqueness of the identity element discussed in the earlier paragraph becomes the stronger statement that if (G,*,-1,e) and (G,*,#,f) are groups with the same underlying set G and the same binary operation *, then their identity elements coincide (e = f) and so do their unary operations (-1 = #).

What both of these solutions lack is the sort of accessibility that would make them suitable for an encyclopedia article that surveys the notion of group. A separate article on group axiomatics which hashed out the different approaches (one-sided, quasigroup, etc.) might be of some interest. But the main article should not dwell too much on axiomatics.

Slawekk's example, by the way, is what semigroup theorists would call a left-zero band. It does indeed show that a right identity axiom and a left inverses axiom do not suffice to define a group. --Michael Kinyon 19:41, 3 August 2006 (UTC)

Closure

The axiom of closure:

(Closure) for all a and b in G, a * b belong to G.

is superfluous, by definition of a binary operation. It's worth mentioning that closure follows from the definition, though.

The test of closure in the examples is in fact a test that the described mapping is inded a binary operation.

Any thoughts before I wade on in and make changes?

I think you should be more precise here. It doesn't follow from the fact that * is a binary operation but that it is a binary operation on G . Is it Ok with everybody if I change that? -- Jan Hidders 08:37, 9 Aug 2004 (UTC)

For the sake of clarity I returned the axiom of closure today.

I am *not* a mathematician, and I found it *quite* confusing, when after reading the definitions section, I found four items listed in the proofs later in the article.

I would make the argument that "group" by definition contains all three of the axioms. The argument over what is an axiom in this case is esoteric in the extreme.

The point is, who are you writing for: (1) an elite group of mathematicians; or (2) a general audience? I would argue that every Wiki article should indeed be rigorous, but at the same time should be clear to as wide an audience as possible. I am convinced that this article overstepped clarity for extreme mathematical exactitude. But this is not a math journal, and is (or should be) aimed at the informed layman, which requires a less formal style.

BTW FYI: To judge how mathematically sophisticated I am, and thus the validity of my argument I think I should disclose that I have a doctorate in biophysics, and as an undergraduate took eight semesters of math including two senior level classes.

Nwbeeson 18:12, 19 January 2007 (UTC)

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"This was our first example of a non-abelian group, because the operation o here is not commutative as the table shows. " If the table did show commutativity, would it be symmetrical about the diagonal from top left to bottom right? TimJ 5 Feb 2002

Yes. The group is abelian if and only if the table is symmetric about the main diagonal. --Zundark, 2002 Feb 5

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Could someone put up a good description of Sylow's Theorem?

See Sylow theorems.

Also, it'd be nice to see a page dedicated to examples of groups.

Nice exercise: Classify all (isomorphism classes of) groups of order <=60. I'd like to see a page on that.

Do you realise how complicated this is, especially for order 32? Doing order <= 15 might be feasible, however. --Zundark, 2002 Feb 22

I started it at list of small groups. AxelBoldt

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It simply is not true that the translation group is "our first example of a Lie group" (as the article is currently arranged). (Z,+) is also a Lie group; it is simply discrete, or 0 dimensional. 0 dimensional Lie groups (discrete groups) are studied in ordinary group theory rather than Lie theory, but they are still technically Lie groups. Hence the necessity for the adjective "nondiscrete". (If you want to change "nondiscrete" to "nontrivial", then I won't fight that, although I won't advocate it either.) -- Toby Bartels, 2002/04/03

Is it common to allow 0-dimensional manifolds? What could possibly be gained by that? Every set is a 0-dimensional manifold. In EDM they don't specify what n is in an n-dimensional manifold, but from their definition of "manifold with boundary" it is clear that they implicitly assume n>0. Furthermore, would you typically find or expect Z_p in a list of simple Lie groups? AxelBoldt

Since the upper half plane of R⁰ is all of R⁰, every 0D manifold with boundary is actually a boundaryless manifold. So it is safe to ignore n = 0 in that case. But I certainly hope that EDM allows for n = 1 for manifolds with boundary. In that case, what dimension is the boundary of this manifold with boundary? If you don't believe in 0D manifolds, you'll have to put an annoying "unless n = 1" into your statement of the theorem that the boundary of an nD manifold with boundary is an (n-1)D boundaryless manifold. And that answers your question "What could possibly be gained by that?" -- you gain the rest of the just at night knowing that you don't have to hunt through your theorems for the odd "unless" and "except". This is just one example of my favourite principle in mathematics: Don't ignore the trivial case! The trivial case may seem uninteresting (that's why we call it "trivial"), but if you slight it, then it will come back to haunt you with a thousand paper cuts.

As for lists of simple Lie groups, I wouldn't expect to find Z_p because the classification of simple discrete groups is part of another subject. (The only things that Lie theorists need to classify are simple connected Lie groups.) But I certainly wouldn't ignore that other subject when classifying semisimple Lie groups (unless I were only interested in the connected ones, again). For example, the Lorentz group is not connected and can't be decomposed into connected simple Lie groups. But it is a semidirect product of a discrete group and a connected semisimple Lie group. So it can be decomposed into simple Lie groups, where some of these simple Lie groups are discrete and some are connected. (The only Lie group that is both discrete and connected, of course, is the trivial group, which is too simple to be simple.) Again we see that the trivial dimension cannot be profitably ignored.

Nobody working on Lie groups is ignorant of their need for the classification of simple discrete groups. But they're so used to the fact that simple Lie groups come in 2 families (the discrete ones and the connected ones) and the fact that only 1 of those families is classified by their own field of mathematics that they naturally consider only their family to be simplie as Lie groups. And this is easy to make rigorously true by redefining the term "simple Lie group". Instead of saying that a simple Lie group is a Lie group that has exactly 2 normal subgroups, say that its normal subgroups fall into exactly 2 dimensions, or that its Lie algebra is simple. (The classification of simple Lie groups, after all, is generally studied on the level of Lie algebras.)

So to sum up: No, I would not expect to find a discrete group on a list of simple Lie groups. But yes, I would expect to find -- and have found -- semisimple Lie groups in use that have nontrivial discrete normal subgroups, and I would expect the Lie theorists that use them to know and understand this. Lie theorists don't study discrete groups, but they ignore them at their peril, and the latter is true of anyone that would ignore the trivial case.

-- Toby Bartels

PS: Could change "nondiscrete" to "connected" as well as "nontrivial", assuming that the trivial group is never discussed above the word, but I think that that gets less at what we want to say. Could also put the word in parentheses to indicate its status as a technicality. -- Toby

I am convinced. Allowing dimension 0 gives a much nicer category of Lie groups. For instance, the kernel of a map between "ordinary" Lie groups may very well end up to be a discrete group. We should probably add this to Lie group. AxelBoldt

That's probably a better argument than anything that I said. Heh. (Certainly related to what I said but better expressed.) -- Toby Bartels

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The article contained a characterization of GL(n,R) as consisting of rotations, reflections, dilations of Rⁿ that keep the origin fixed; SL(n,R) was likewise explained as the rotations and reflections. I added those originally, then they were removed and readded. But they are indeed wrong. Even SL(2,R) contains lots of transformation different from rotations and reflections. There are skew transformations like [[1 1] [0 1]], dilations like [[2 0] [0 1/2]], and combinations of these like [[2 1] [0 1/2]]. AxelBoldt, Thursday, May 23, 2002

I put them back because I thought that I saw the error — you didn't specify that the origin was fixed, so affine transformations were included. Of course, the errors were more extensive, so I've now gone over everything more carefully. What we have now should be correct; I hope that you like it. — Toby Bartels, Thursday, May 23, 2002

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Axel, you say that there are elements of SO(n) that are not rotations. Did you have an example in mind? I'm 100% certain that this is wrong, and I checked with 2 other mathematical physicists (Miguel and John Baez if you want credentials), who agree that I am not having an acid flashback or something. (Heck, it's even called the "rotation group" — which would make a nice name for an article on it.) I'm not sure that I like the description of GL(n) yet, and I'll think about that some more, but in any case, surely we can agree that the rotation group consists precisely of rotations? — Toby Bartels, Tuesday, May 28, 2002

The example I had in mind is reflection at the origin in R⁴: [[-1 0 0 0] [0 -1 0 0] [0 0 -1 0] [0 0 0 -1]]. Or can that be interpreted as a rotation somehow? I don't even know what a rotation is :-) AxelBoldt, Tuesday, May 28, 2002

When I talked to John, he remarked "How obvious it is depends on how you define a rotation. I define it as an element of SU(n).". Your matrix has infinitesimal generator (say) a := [[0 1 0 0] [−1 0 0 0] [0 0 0 1] [0 0 −1 0]]; we can transform R⁴ continuously (even smoothly — analytically if you get right down to it ^_^) along the path t → exp(ta); at each stage, globally and infinitesimally, we preserve lengths and angles; at t = π, we reach your matrix. (Of course, there are all kinds of curvier paths that we could take, but I think that this is a minimal geodesic, so it must ^_^ be best.) This seems intuitively like a rotation to me. (And this talk of preserving lengths and angles isn't entirely begging the question — we get orientation preserving for free. The same thing shows why only the proper orthochronous part of SO(3,1) is the Lorentz group of rotations on Minkowski spacetime.) I don't know whether or not this cuts it as an intuitive argument, but I'm still certain that both mathematicians and physicists would count your matrix as a rotation — and who besides them ever thinks about rotations in R⁴ anyway? -- Toby Bartels, Friday, June 7, 2002

I like this notion of rotation. I had some muddy image in mind where you stake a line through the space and then rotate around that line, but this is a lot cleaner. I will officially give up my resistance to the term "rotation" and retract everything I said before. AxelBoldt, Friday, June 7, 2002

Ah, but in 4D a rotation is about a plane... :) BTW, has anyone written about Jean-Pierre Petit? He does great vulgarization of maths in comic book form, his book on topology (which covers Klein bottles & Boy's surface etc) has a preface which recommends readers have a packet of aspirins & a glass of water to hand... same applies here I think ;) Tarquin

And in 2D, a rotation is about a point. Also, in 4D, a rotation need not be about a single plane (Axel's −1 is an example of such) but may instead be around a sort of linear combination of planes. This is related to the fact that not every 2form in 4D can be factored as a wedge product of 1forms.

In general, the direction of the "axis" of a rotation in nD can be described by an (n−2)form, specified up to a scalar multiple. This (n−2)form is another way of looking at the infinitesimal generator of a rotation. You can see how the infinitesimal generator that I gave for Axel's rotation is a linear combination of 2 blocks, which correrspond to 2 factorable 2-forms and hence 2 planes. The rotation is about both of those planes simultaneously. -- Toby Bartels, Sunday, June 9, 2002

So it is true that any element of SO(n) is a product of rotations about n-2 dimensional hyperplanes? AxelBoldt

I'm pretty sure that that's true. If it isn't, then the analogy between axes of rotation and 2forms doesn't work like I think that it does (but perhaps that's exactly what you're getting at). I can check it out if you want. — Toby Bartels, Tuesday, June 11, 2002

I'm not trying to get at anything, just trying to learn. I believe it is true myself, using the real version of the spectral theorem, which says that to every element A of SO(n) there's an orthogonal S such that SAS^t is a matrix with blocks of the form [[cos x -sin x] [sin x cos x]] (and a couple of ones) on the main diagonal. The blocks probably correspond to the hyperplane rotations. AxelBoldt, Tuesday, June 11, 2002

That sounds about right to me. — Toby Bartels, Tuesday, June 11, 2002

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This page has gone back and forth multiple times on the reformatting of "Examples and counterexamples". Some thoughts:

First, mathematically speaking, the integers, under multiplication, is not a "counterexample" of a group; it is simply not an example of a group. (Nor is the set of all long-haired dogs a "counterexample" of a group). So we could change that to "Examples (and non-examples)". But that seems silly; we don't generally explicitly provide "non-examples" as part of other articles in the Wikipedia World of Mathematics, except in passing.

This is simply not true. It's always important to provide examples of what something is not as well as what something is. I've written a lot of material on various kinds of topological spaces (you can find most of them linked from Separation axiom), and I always include nonexamples, as Axel can attest. I agree with you that "counterexample" is not the right word, however.

Sheepish withdrawl of previous statement; but although I agree that we want to have "non-examples" from a pedagogical view, I find the terminology a bit clumsy here. One thing I note by looking through the examples that you cite is that, although we can talk about non-normal spaces and non-paracompact spaces, etc., we don't talk about topological non-spaces; similarly, although we talk about abelian groups and non-abelian groups, solvalable and non-solvable groups, etc., we don't talk about non-groups.

Personally, I find it better to think about an example of something that is not a group (in this case in the context of integers), then it is to think of a non-example of a group, or a counterexample of a group.

All this aside, I think the whole article looks much, much better now - and if you still feel strongly, feel free to change it back to "examples and non-examples"; I've made my last edit on that topic. Cheers Chas zzz brown 02:29 Nov 4, 2002 (UTC)

Secondly, some of these examples (Symmetry in particular) are quite long and wordy; and others (free group) are described in separate articles and at any rate are somewhat unsuitable as part of an introduction; if you don't know that the reals form a group under addition, you're unlikely to grasp the concept of a free group as an "example" of a group.

You definitely have a point here. The article as it stands is unwieldy, and this is part of the reason. We should have some simple examples and nonexamples in the article to reinforce the definition, then move the rest to Examples of groups or some such thing.

Thirdly, I think that, as an introduction to group concepts (which really are fundamental!), we want the article to give give the reader the following information:

• A good general feeling of what constitutes a group
• Enough examples to show that groups are indeed "fundamental" in the sense that an understanding of them progvides insight into many other mathematical structures.
• Some feeling of the historical development of group theory.
• Applications of group theory with links to more info.
• Some fundamental properties of groups that act as a launch pad to link to other info.
• groups as a subset of abstract algebra as a subset of homological algebra as a subset of category theory (generalizations)

Some of this belongs on Group theory, but you have some good ideas. The page definitely needs to be redesigned.

Right now, it seems like the formatting issue of proceeding with a series of examples, with everything else thrown in at the end, is keeping the article from being as useful as it could be; at the minimum, I'm changing "Examples and Counterexamples" to "Examples". Any thoughts? Chas zzz brown 20:51 Nov 2, 2002 (UTC)

— Toby 15:13 Nov 3, 2002 (UTC)

How about if we move (most of) the examples out of the way and put them on Examples of groups? AxelBoldt 21:59 Nov 2, 2002 (UTC)

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The article is growing too long. Should rewrite it. I will try to mimic the style as in topology, topology glossary and topological space. Wshun

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I'm wondering whether someone more familiar with this material could revise the beginning of this article. Saying that the concept of "group" is fundamental to modern algebra says nothing about what it is. Admittedly we don't need to go far in the article before we encounter basic definitions, but these will be of no help to a person who has difficulty following these very elementary mathematical expressions. I got here when I tried to deal with the word "Abelian" in Wiktionary. Although it is very easy to deal with the concept of Abelian group once a person knows what a group is, getting from set to group has not been made so obvious. ☮ Eclecticology 20:36, 2003 Sep 8 (UTC)

You're basically correct; however, this is about the hardest part to write in an article of this sort. I certainly don't have a good idea of how to write a helpful introduction. But you are right that we could use one! -- Toby Bartels 14:00, 16 Sep 2003 (UTC)

Hope this helps -- Daran 17:00, 16 Sep 2003 (UTC)

Proofs

I think it could be potentially useful to include proofs in the Wikipedia of many of the simple (and more complicated) theorems listed here. However, I feel at a loss for how to format such an endeavor. It's silly to create a page for a trivial theorem, and it disturbs the flow too much to insert it into this page. Perhaps a page should be made containing proofs of a series of simple related theorems? This seems to defeat searching and indexing in some respects. What are your thoughts?

I think this is a good idea. Very short proofs can be included in the text itself, slightly longer proofs can go in their own section, particularly long proofs can go in their own article, like Sylow theorems. In the middle is proofs too big for a section but too small for an article. In this case, as you suggest, I think the best thing is to have an article collecting several related proofs and linking it from the relevent pages. This is particularly useful for proofs relating several topics. Deco 18:27, 14 July 2005 (UTC)

Nonempty

Aren't groups supposed to be nonempty? This isn't a ridiculous question to ask. For instance, if the empty set is considered a group, then the trivial group is no longer an initial object in the category of all groups, and it seems intuitively clear that the trivial group should be both terminal AND initial. A similar thing does not happen in category of top spaces with continuous maps, though, since here, the singleton space is not an initial object to begin with. Revolver 23:03, 23 Nov 2004 (UTC)

groups are automatically non-empty since they must contain e, the identity element. MarSch 17:50, 26 Mar 2005 (UTC)

Socks and shoes?

What does the 'Socks and shoes' remark mean under #Simple theorems? Shai-kun 13:59, 1 November 2005 (UTC)

I would be interested to know that too Manipe 23:26, 25 December 2005 (UTC)

As far as I can tell, all it's saying is that when taking off your socks and shoes, you remove them in the opposite order from the order you put them on in. It's just a mnemonic name for the lemma. Deco 02:20, 26 December 2005 (UTC)

The neutral element?

What is the symbol e in the third axiom of the definition? It says "where e is the neutral element from the previous axiom" but the previous axiom only postulates that the set of neutral elements in a group is not empty. It does not define any specific neutral element that can be referred to. Slawekk 16:50, 14 March 2006 (UTC)

It can be shown that there cannot be more than one neutral element as defined in axiom two. Assume that e1 and e2 are neutral elements from axiom 2.

• Then by definition e1 * a = a for every a, choosing a = e2 yields e1 * e2 = e2
• Also a * e2 = a for every a, where choosing a = e1 yields e1 * e2 = e1

Combining these results shows that e1 = e2. So all neutral elements are equal, or, in other words, there is only one neutral element. −Woodstone 19:26, 14 March 2006 (UTC)

What you say is true, but does not address the problem. The third axiom references the neutral element saying that it is defined in the previous axiom, while the previous axiom only says that the set of neutral elements in a group is not empty. I know that it all works out in the end and that this definition is repeated in many books. It still does not make it correct. One correct way is to define a monoid with the first two axioms, then prove that there is only exactly one neutral element (which allows to refer to it as the neutral element), and then define group adding the third axiom.

There is no need to give a name to the "thing" defined by the first two axioms. Inserting a note that "the" neutral element is unique based on the first two is enough to proceed into axiom 3. −Woodstone 21:35, 14 March 2006 (UTC)

additive and multiplicative groups

What are the definitions of additive and multiplicative group and how can we recognize that a given group is multiplicative or additive? Slawekk 21:58, 17 March 2006 (UTC)

An additive group is a group that's written additively, and a multiplicative group is a group that's written multiplicatively. So there's no real difference, just a difference of notation. --Zundark 22:56, 17 March 2006 (UTC)

That means that there is no such thing as "additive group" or "multiplicative group", only additive or multiplicative notation? In such case the group (mathematics) entry should not use the notions of multiplicative or additive group as in "The neutral element is usually called the identity element for a multiplicative group and the null element or zero element for an additive group." Slawekk 00:03, 18 March 2006 (UTC)

Well, the particular statement you quote is misleading, as the term "identity element" isn't restricted to multiplicative notation, and while the identity element of an additive group is often referred to as "zero" (since it's written 0), it isn't a zero element. But there's nothing wrong in principle with using the terms "multiplicative group" and "additive group", which are more succinct than "group that is being written with multiplicative notation", or "group that we are writing additively", or whatever. --Zundark 09:23, 18 March 2006 (UTC)

Conventionally abelian groups are written additively where as non-abelian groups are written multiplicatively. LkNsngth (talk) 03:07, 13 April 2008 (UTC)

Elements and terms

>The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non-identity terms.

"Elements" suggests set membership. What is the set the elements from the above sentence belong to? I think they should be called factors of the product rather than elements. The term "terms" is heavily overloaded in different areas of mathematics and computing science. It is better to talk about elements of a group instead of terms of a group. I think non-identity word is redundant here. Each factor in the product contains only one identity element, so it has finite number of non-identity elements iff it has finite number of elements (i.e. is finite). So, the direct external sum of a family of groups is the subgroup of the product constituted by factors that are finite groups. Am I missing something?

You have completely misunderstood it. It's talking about elements of the product, not factors of the product. Perhaps it would be clearer if it said "coordinates" rather than "terms". --Zundark 16:50, 4 April 2006 (UTC)

Yes, it would definitely be clearer. I have never heard of calling the value of a projection map at $x$ a "term" of $x$. Slawekk

identity elements

What is the purpose of an identity element, in any system, if it's the same as doing nothing to the elements, and where did the idea originate? phyti jun 3 06

The identity element e of an operator ^ is the stepping stone to define the inverse of elements. The inverse ā of an element a is the one which satisfies ā ^ a = e. If the operator is addition on numbers, the identity element is 0 and the inverse of a is -a. For the multiplication operator the identity element is 1 and the inverse is 1/a (unless a=0). Hope this satisfies your curiosity. −Woodstone 10:04, 4 June 2006 (UTC)

I know the definitions, that is not the question. Using algebra for instance, we know 1/a is an inverse/reciprocal by defintion and +a-a=0. One and zero are already defined. When would you ever add 0 to a number, or multiply by 1 when you know what the answer is, no change. I.e. you can still perform all operations without defining an identity element (this does not mean we discard 0 or 1). It seems redundant. phyti-jun 4 2006

I'm not sure what you're aiming at, but it is not possible to define a group without defining what a neutral element is and postulating that there is one.

As for adding 0 to something, that happens routinely in all kinds of calculations, especially when executed in a computer. It is simpler to just add and get the right result, than to check if one of summands is 0 first. Similarly in an abstract formula it is much easier not to make a special (exception) statement for zero, when it could ride along in the general case. −Woodstone 20:07, 4 June 2006 (UTC)

Thanks, I guess I'm wishing for a perfect system and there isn't any.phyti-jun 05 2006

The identity element isn't intended for computation; you wouldn't go around multiplying numbers by 1. As Woodstone says, it's a stepping stone for defining other important concepts and proving results, and a convenient notational shorthand for expressing the idea of a fixed point. For example, you can interpret the statement "a*a^-1 = e" as meaning "if I multiply something by a^-1, then by a, I achieve the same effect as though I did nothing". The proof of the lemma "(ab)^-1 = b^-1a^-1" depends critically on e although it does not contain e in its result. Deco 23:24, 5 June 2006 (UTC)

Also, another thing about the identity element is that it plays a significant role in defining functions and/or figuring out what functions do. for example, you can tell a lot about a function just by seeing what it does to the identity element. The identity is also necessary to keep a group closed under operations (it just comes naturally from the definition)... for example, if you had no identity element, you wouldn't be able to compute things that are very trivial, like for example, aaa^-1b, as you cannot assume that aa^-1 is just 1. Spindled 01:59, 20 October 2006 (UTC)

And a good entry is gone ...

On October 20th 2006 Spindled stopped by and erased lots of important information in this entry, contributed mostly by Michael Kinyon. This turned an informative and interesting entry which had been a result of the discussions above into a mediocre one, on a high-school popular level. Slawekk 16:22, 20 October 2006 (UTC)

Can you be more specific as to which important information I have erased? All I've done was reword the definition in a simpler way and clarified some information on order, notation and proof methods. I have erased the table for notation as it was quite limiting in stating different notation for permutation groups and function groups, but all the information is still there. Spindled 22:34, 20 October 2006 (UTC)

Quite a lot has been removed

1. The concept of closure. While technically not group axiom, usually listed and very important.
2. Use of one particular notation (composition) in the definition, which could lead to some confusion.
3. The standard notion for writing a group (S,+).
4. The logically minimal alternative axiomation.
5. Commutative groups.

I'm afraid I'm going to revert the changes as too much has been lost. --Salix alba (talk) 22:55, 20 October 2006 (UTC)

I think reversion is justified as a lot had been deleted, but I think there were some good things in the reverted content. I'm going to try to merge some of it together. - grubber 23:51, 20 October 2006 (UTC)

I think that some of the notation and terms used here are non-standard, and hence why i removed them. The term 'neutral element' as far as i know is not standard (as the identity element also holds more information than just being neutral, it also stresses that the element gives us important information about the group). The things about left and right neutral elements is more confusing than helpful for beginners (and unnecessary for non-beginners) IMO. I also initially moved a lot of the stuff from the definition to the notation part (without deletion). In any case, i propose my previous edit with two section regarding the identity and inverses (I have some prepared). I think the wording in general needs to be simplified a bit. Another thing was regarding the 'alternative axiomization'. Although it might be sufficient to just show right neutrality and right inverses, it is usually required to show both when proving closure for inverses and identity - by definition (I have 2 different sources - It is also less rigorous to just show one side). I'm going to give it another shot. If you still don't like it then let me know. Spindled 02:38, 21 October 2006 (UTC)

Ok i'm finished. Let me know if its any better than the first. I might've included too many examples of identities and inverses in their sections, if so feel free to reduce them. Spindled 03:46, 21 October 2006 (UTC)

There's still some work to be done here. - grubber 04:49, 21 October 2006 (UTC)

1. "A group (G, *) is a non-empty set" - "non-empty" is reduntant, it follows from the second axiom in the definition.
2. " There is an element e in G such that for all a in G, e * a = a * e = a. For each a in G, there is an element b in G such that a * b = b * a = e, where e is the neutral element from the previous axiom" - It is incorrect to refer to a variable bound by the existential quantifier outside of the scope of that quantifier. This means the if you say "$\exists _{e\in G}\phi (x)$" (like in the second axiom), then you shouldn't say "where e is the the neutral element from the previous axiom" (see also the discussion above). This incorrect wording was present in the pre-Oct 20 versions of this article, because it is (unfortunately) very common in textbooks. However, there was a remark ending with "This is why the third group axiom refers to the neutral element, even though the second axiom merely asserts that there is at least one neutral element.", which corrected this problem and this remark is now gone. As for calling the neutral element "the identity element" it is OK to use any convention, as long as the usage is consistent (it is not now) and the article mentions the alternative terminology.
3. The missing paragraph on alternative axiomatizations: "The things about left and right neutral elements is more confusing than helpful for beginners (and unnecessary for non-beginners)" - You seem to assume that the readers are either beginners who know nothing or experts who know everything. I think most readers are neither. I think an article should be structured so that it starts with a general overview accessible for everybody, then proceeds to a more formal definition and covers more and more specialized material in later sections for more advanced users. At some point there should be some material that beginners will not be able to immediately understand (is "confusing" for them, as you call it). This way we know that the article does not target only the beginners. I found the section on alternative axomatizations very interesting. "Although it might be sufficient to just show right neutrality and right inverses, it is usually required to show both when proving closure for inverses and identity - by definition" . I am not sure what you mean by "it is usually required" here. If you mean "it is usually required when doing a homework exercise in an abstract algebra class", then you are right as most textbooks don't use the alternative aximatizations described in this section. Otherwise, what is required depends on the definition used. Perhaps the original section should be titled "Equivalent definitions" rather that "Alternative axiomatizations".
4. The "Notation for groups" section: in the pre-Oct 20 versions the wording of the article was chosen carefully to avoid using the expressions "additive groups" and "multiplicative groups". These expresions make people think that "additive groups" and "multiplicative groups" are separate mathematical notions and there is some fundamental difference between groups for which the operation is written using the additive notation and those for which it is customary to use the multiplicative notation. This misunderstanding is common among physicists, for example. Slawekk 17:15, 23 October 2006 (UTC)

The identity element will always commute with all other elements, so i thought there was no reason to mention a "right neutral element" or a "left neutral element". As for referencing the identity element in the second axiom, i didn't suggest that, but it only clarifies what the e stands for in the third axiom, it does not depend on it. As for showing both sides in a proof, this is usually necessary to actually prove that the right and left identities are in fact the same, which is implicit in the definition (ae=ea=a uses e on both sides.) One factoid we might want to consider including is that e is always in the centre of G, which might clarify some things. As for the notation part, multiplicative groups and additive groups do have some significant differences in terms of notation (e.g. the identity is a different element in each, you have to multiply instead of add, etc.) but it might use a different wording i guess... maybe a clarification that indeed there is no fundamental difference. Spindled 19:40, 23 October 2006 (UTC)

I have restructured the Definition section to take care of most of your concerns. I removed most of the "example" material, since that is all done later. I'm not happy with the "Notation" section, and I will tackle that section next unless someone beats me to it. - grubber 19:58, 23 October 2006 (UTC)

I think the new version is clearer, especially the briefer definitions and axioms. I have never seen the term 'neutral element' before so I have moved it down to the notation section, and made a few other minor edits. But I dont like the clock. 'This picture illustrates how the hours in a clock form a group' - no it doesn't, what group is it supposed to be? Z_12? SO(2)? Either this needs more explanation or it should be dropped. grubber - I have made some changes to the notation section eg making it less dogmatic (is -> may) Paul Matthews 13:35, 31 October 2006 (UTC)

Great edits. I think this page is getting better. If you want to work some more on groups, check out Group theory. I added a section on group concepts there, minus the rigor and detail. I'm still trying to figure out what the distinction between that page and this page should be. Your help over there would be appreciated! (And I agree about the clock. It could even represent the angles 0 to 2π. I wouldn't mind seeing it go. It would be nice to have some interesting image, maybe representing a dihedral group or something). - grubber 16:03, 31 October 2006 (UTC)