# Talk:Group (mathematics)

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## Division?

The generalisation section says: For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. That seems clear enough, but when I look at the accompanying table I see that groups and monoids differ by "division". Do we mean "inverse" instead? Whatever we mean the table needs to be clearer and match the text. --Michael C. Price talk 08:39, 16 May 2009 (UTC)

I've changed the table to clarify this (see Template talk:Group-like structures#'Divisibility' changed to 'invertibility') and have renamed and rewritten the 'Division' section as well: Group (mathematics)#Invertibility of the group operation. It only took seven years! splintax (talk) 10:46, 28 September 2016 (UTC)

## Blue element in figure 'Group table of D4'

In the figure 'Group table of D4', there are various sections coloured with different concepts related to the group. There is one colour - blue - which covers the element fh•rd, but this is not explained in the caption. What concept does the blue colour represent here? --Fearedinlasvegas (talk) 13:33, 8 December 2016 (UTC)

As explained in the text, it is just highlighted to emphasize the order of the product. — Preceding unsigned comment added by Jakob.scholbach (talkcontribs) 18:07, 8 December 2016 (UTC)

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## Section Request

This page should discuss some more tools for constructing groups and give specific examples. This should include

• Finding subgroups of ${\displaystyle S_{n}}$  using a finite set of conjugations
• Constructing groups as limits and colimits
• Such as the additive group of ${\displaystyle p}$ -adic integers
• Construction of ${\displaystyle SL_{2}(\mathbb {Z} )}$  from ${\displaystyle \mathbb {Z} /2*\mathbb {Z} /3}$
• Semi-direct products using matrices

— Preceding unsigned comment added by 74.220.45.91 (talk) 20:51, 30 November 2017 (UTC)

## Group homomorphisms - Cayley's theorem

The last paragraph of the Group homomorphisms section describes Cayley's theorem applied to finite groups:

One useful theorem about finite groups is that every finite group admits an injective group homomorphism into the symmetric group of a finite set. This can be constructed by taking the finite set as the underlying set of the group {\displaystyle G} G and sending a {\displaystyle g\in G} g\in G to the set-automorphism given by {\displaystyle g\cdot } {\displaystyle g\cdot }.

This paragraph seems out of place given the level of the rest of the article. If we want to keep it, we should find a way to re-write it to be as readable to the lay-person as the surround text. I also think that (if we keep this), we should add an explicit mention and link to Cayley's theorem and remove the restriction to finite groups. Having said that, ignoring the readability issues, Cayley's theorem seems like it is still a bit to advanced to warrent inclusion in an otherwise 2 paragraph long introduction to homomorphisms. Homura1650 (talk) 07:23, 5 December 2017 (UTC)

I agree with you. The edit was certainly done in a constructive mood, but we do mention Cayleys theorem in the § about finite groups (where it belongs). I don't think a more detailed explanation why Cayleys theorem holds is in order there.
I suggest reverting the edit you quote. Jakob.scholbach (talk) 10:02, 5 December 2017 (UTC)

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## Finite fields

I have twice reverted the inclusion of Finite field as the main article for the section on modular arithmetic. It is true that some finite fields are of this type, but not all! When the modulus is a proper power of a prime, the field of that order and modular arithmetic are not related. Furthermore, the modulus in modular arithmetic is not limited to primes or powers of primes, so again there is no general correspondence with finite fields. Indicating that finite fields are the main sources of modular arithmetic is just plain wrong! --Bill Cherowitzo (talk) 18:50, 26 June 2018 (UTC)

1) text on this part of page is too short;
2) text on this part of page is similar to part "Finite Field" of page "Field", the picture is same;
3) can You (Bill Cherowitzo) redirect this part of Page "Group" to page "Finite Group" ?
With the hope for a constructive solution,
   Grigory Bilchenko

Ggbil2 (talk) 19:42, 26 June 2018 (UTC)
P.S. The first part of the page "Finite Ring" is named ... "Finite Field" ;)
Ggbil2 (talk) 20:00, 26 June 2018 (UTC)
I agree with Bill Cherowitzo. Also, according to Template:Main, because subsections are often written in summary style, this template is used after the heading of the summary, to link to the subtopic article that has been summarized. In this case the template is not appropriate, because the subsection is specifically about modular arithmetic—not finite fields. There is currently no part of the Group (mathematics) page that should be redirected to the Finite field page. However, the Modular arithmetic subsection already links to the Modular arithmetic article.—Anita5192 (talk) 21:04, 26 June 2018 (UTC)

## Definition (again)

the requirements ea = a = ae and a -1 a = e = a a -1 are redundant. sufficient is: ea=a and a -1 a = e. This is posed as an exercise in Group Theory .. by Norton Hamermesh. With the reduced definition one can prove that none of the axioms is redundant (id est, none of the axioms can be proved by the other axioms) Should this be included in the article? Jacob.Koot (talk) 16:07, 28 September 2018 (UTC)

This is already mentioned in Elementary consequences of the group axioms.—Anita5192 (talk) 16:51, 28 September 2018 (UTC)
My mistake, sorry. Nevertheless, I don't see mention of the fact that it can be proven that none of the axioms can be omitted. Also that the identity e in ea=a must be the same as the identity e in a -1 a = e, which is essential too. Jacob.Koot (talk) 17:17, 28 September 2018 (UTC)
If any of these elementary principles are not in the article, you may want to include them.—Anita5192 (talk) 17:21, 28 September 2018 (UTC)

It can be proven that none of the axioms can be omitted and that it is essential that the identity e in axiom

ea = a

is the same one as identity e in the axiom

a-1a = e

Jacob.Koot (talk) 18:08, 28 September 2018 (UTC)

First of all, this is obvious, since the same variable, e, is used in both equations. Second, I avoid using the word one in mathematical discussions, as it can be ambiguous. I would have said, "is the same identity as e in the axiom . . ." Third, I think it is more important to state in general that the identity e is unique, which is easy to prove: assume the existence of two identities, e1 and e2, and then prove e1 = e2.—Anita5192 (talk) 19:02, 28 September 2018 (UTC)
My point is that it is essential that in both axioms e is the same. It is possible to construct a system that satisfies all axioms except that the (or a) identity is not the same in both cases. For example:
 e e' e' e'

It satisfies all axioms, except that it has two distinct identities and is not showing a group. I think mentioning that the two identies must be one and the same is essential. This is clear from the definition, but I think it should be mentioned two distinct identities not necessarly produce a group. Consider it as part of showing that the axioms cannot be reduced to less restrictive axioms. Jacob.Koot (talk) 19:19, 28 September 2018 (UTC)

Please look at your example again. Your element e' is not an identity unless e' = e (since e' ⋅ e = e'). If all the group axioms are satisfied, you can not have two distinct identities, as Anita5192 has pointed out. In fact, this can be shown in much weaker systems. A groupoid having a left identity and a right identity has a unique (two sided) identity. Most authors would not consider it necessary to stipulate possibly different identity elements in the statement of the axioms, since it is completely trivial that they would have to be the same and this statement would only lead to unnecessary confusion. --Bill Cherowitzo (talk) 20:29, 28 September 2018 (UTC)

Look at it as follows:
1: forall x,y in G : xy in G (composition)
2: the composition is associative
3: exists e in G : forall x in G : ex = x (left identity)
4: exists e' in G : forall x in G : exists y in G : yx = e'
5: e' = e

Axiom 5 cannot be omitted.
There are systems (not groups) that satisfy 1, 2, 3 and 4 but do not satisfy 5. See my example above.
Hence axiom 5 is necessary.
Notice that I left out the right identity.
Its existence can be proven as well that it necessarily is the same as the left identity. Jacob.Koot (talk) 11:00, 29 September 2018 (UTC)

What you just defined is not a group.
However, looking at the definition in the article, I noticed several deficiencies: 1. the operation should be referred to specifically as a binary operation. 2. the result, ab, should be specified as unique, that is, the binary operation is well-defined as well as simply defined. 3. the line under Identity element indicating that "Such an element is unique . . ." should be removed, as this is a result of the definition—not part of the definition. If nobody has any objections, I will make these changes.—Anita5192 (talk) 16:30, 29 September 2018 (UTC)
More simply stated, having 4 in this form is convoluted. This axiom defines the inverse, not a new unity and an inverse. It should simply say:
4: forall x in G : exists y in G : yx = e
where the e is implicitly the e of the previous line and 5 becomes meaningless. −Woodstone (talk) 18:05, 29 September 2018 (UTC)
I agree. Your #4 does not define a right inverse for each element unless e' = e. A semigroup with a left identity is a group if and only if each element has at least one right inverse (Bruck, 1971). Your approach is being overly convoluted and I am not seeing any advantage to it. --Bill Cherowitzo (talk) 19:08, 29 September 2018 (UTC)

The point I wanted to make is that in the two axioms
forall x in G: ex = x
forall x in G: exists y in G: yx = e
it is essential that both axioms have the same e.
Just to show that the axioms cannot be simplified. It is not very difficult to prove that no part of the axioms can be omitted, the rule that e must be the same in the two axioms above included. That is my point. Jacob.Koot (talk) 15:21, 30 September 2018 (UTC)

What you are saying is clear to me, but I do not believe that it is making the point you think it is making. The problem is that your axiom 4 is not a group axiom as you have stated it. Let me try to expand on this. Call a system that satisfies your axioms 1-4 a Beastie. You have a theorem that says a Beastie is a group if and only if the statement of axiom 5 holds. Every group is a Beastie, but there are (proper) Beasties that are not groups (as your example shows). A Beastie is a generalization of a group. Now we have to leave the realm of mathematics and ask, is the concept of a Beastie useful? Is it interesting in some way? There are no mathematical answers to those questions as they are based on subjective judgments. I would say that they are not since proper Beasties are not quasigroups (do not have multiplication tables that are Latin squares) and can not be embedded in any group (although they could contain a group as a substructure). Others may view this differently and find them interesting, but it is clear that they are not groups nor even structures that are in some sense precursors of groups. Your claim that this theorem about Beasties shows that axiom 5 is essential as a group axiom does not follow. Take any theorem whose conclusion was that something is a group and any hypothesis of that theorem. By weakening the hypothesis you can lose the conclusion and in this situation, anything that re-strengthens the hypothesis, you would call an essential axiom in the definition of a group. This is the argument you are making in this specific case and when looked at from a general viewpoint you can see that it is not convincing.--Bill Cherowitzo (talk) 19:15, 30 September 2018 (UTC)
It convinces me. BTW, I am not suggesting to add axiom 5 as a separate axiom.
I only like to emphasize that in the axioms ex=x and exists y: yx=e, the same element e is essential.
Consider the set {0, 1, 2, 3, 4} with associative (even abelean) composition f(x,y) = min(x+y,4).
We have f(0,x)=x and f(4,x)=4, but do not have a group here. It is a Beastie, as you call it. Jacob.Koot (talk) 11:49, 1 October 2018 (UTC)

## Switch to LaTeX?

I've been switching Wikipedia math pages to LaTeX to aid presentability, but I've been advised that I should write on the talk pages before doing so.

I think this article should render math equations via LaTeX. It currently is mostly not using LaTeX, which is surprising because this is an extremely important article. I can do this in a pretty systematic way using Visual Code Studio.

Proposed version on the left; ugly current version on the right.

My reasons:

1) You don't have to do anything and only the math symbols will be changed.
2) TeX can be rendered on any modern browser and mobile device. I checked so myself.
3) TeX is easier to type out. Compare ''F'' : ''C''<sup>2</sup> → ''D''<sup>2</sup> which produces F : C2D2 with :$F:C^2 \to D^2$ which produces ${\displaystyle F:C^{2}\to D^{2}}$ .
4) The math rendering using apostrophes, for example ''F'' : ''C''<sup>2</sup> → ''D''<sup>2</sup> which creates F : C2D2, is extremely limited. For example, how are you going to create an integral? A fraction? Are you going to use some kind of ridiculous unicode symbol?
5) TeX looks nicer, and presentability is extremely important for math pedagogy. Modern mathematicians are starting to wake up and realize that we need graphs, figures, diagrams, colors, and better formatting to explain concepts and to overall write better books. The ultimate goal is to help people learn these concepts, and right now this article suffers from styling issues which I believe can make it harder to understand.
6) Overall, I think anyone will be sort of turned off when visiting this article and discovering that the math has been formatted in a horrible, ugly and unreadable manner, and will then go somewhere else to read about groups.

An example of my proposed changes can be compared side by side; see the thumb pic on the right. If nobody has any objections/no one responds, then I will go ahead and implement the changes since it's nothing risky anyways (as far I have researched anyways; correct me if I'm wrong). — Preceding unsigned comment added by Lltrujello (talkcontribs) 23:38, 17 March 2020 (UTC)

Generally no. I do usually support change to {{math}}/{{mvar}} for simple, inline stuff, and $...$ is generally okay for more complicated inline stuff and anything that's set apart on its own line, but changing every little bit of math on a page to $...$ mode isn't ideal. It tends to hinder accessibility; you can't copy it to the clipboard; etc. –Deacon Vorbis (carbon • videos) 23:42, 17 March 2020 (UTC)
Could you provide different reasons? I'm not sure what you mean by accessibility. As I pointed out, TeX can be rendered on modern web browsers and mobile devices. I also think it'd be more readable for someone with an eye disability; for example, as it is now. It is difficult to distinguish math characters with regular words because they're so closely similar.
Also, I'm not sure what you mean about copying and pasting. I'm a workaholic math student, so I read math notes, books, and surf wikipedia math articles and stack exchange sites every day. But I cannot remember a single time I've ever wanted to copy and paste a math expression. The only possible reason I'd want to do that would be to google something, but it's generally not fruitful to google specific unicode characters. Lltrujello (talk) 02:31, 18 March 2020 (UTC)
• Full mathematics formatting is needed for formulas in the "Rationals", "Galois groups", "Finite groups", and "Topological groups" subsections. My position is that we should not mix and match formatting: a single style should be used consistently within any single mathematics article, because otherwise we get the same thing formatted in different ways, unnecessarily confusing the readers. So because we need LaTeX somewhere in the article, I am in favor of using it everywhere in the article. —David Eppstein (talk) 23:48, 17 March 2020 (UTC)
Oppose The use of $ . . . $ tags is useful for complicated mathematics typesetting, for example, integrals and fractions, but I disagree with (3) above. For simple inline variables like a, b, etc., apostrophes are much easier and less error-prone. And retroactively changing everything is a lot of work to do, a lot of edits for other editors to check, and a lot of opportunities for mistakes. We have more important editing to do. — Anita5192 (talk) 00:00, 18 March 2020 (UTC)
• Let me give you a better example. Observe that "If {{nowrap|1=''f'' : ''a'' → ''b''}}, {{nowrap|1=''g'' : ''b'' → ''c''}} and {{nowrap|1=''h'' : ''c'' → ''d''}} then {{nowrap|1=''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f''}}" (From the Category Theory page) which creates
if f : ab, g : bc and h : cd then h ∘ (gf) = (hg) ∘ f
can be compared with "If f: a \to b, g: b \to c and h: c \to d then h \circ (g \circ f) = (h \circ g)\circ f" which produces
if ${\displaystyle f:a\to b,g:b\to c}$  and ${\displaystyle h:c\to d}$  then ${\displaystyle h\circ (g\circ f)=(h\circ g)\circ f}$
Clearly, the previous expression with the ridiculous apostrophes is extremely time consuming to write and error prone. I don't see how anyone could possibly believe the opposite. Furthermore, the previous equation requires 178 characters with 192 bytes while the LaTeX-ed equation requires only 83 characters and 83 bytes. I've come across many expressions like these, and have felt sorry for the previous authors who probably spent a few minutes on an equation which could have been written in a few seconds with LaTeX.
Ultimately, the goal here is to make mathematics readable. I'm trying to make math better for other people just like everyone else on these sites. But times are changing, LaTeX is becoming more widespread, and the novice math people, which this article targets, are not going to like the hard-to-read, apostrophe-written equations that are here, and will prefer LaTeX. Lltrujello (talk) 02:31, 18 March 2020 (UTC)

"If {{math|1=''f'':''a'' → ''b''}}, {{math|1=''g'':''b'' → ''c''}} and {{math|1=''h'':''c'' → ''d''}} then {{math|1=''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f''}}" creates

"If f:ab, g:bc and h:cd then h ∘ (gf) = (hg) ∘ f

Using {{math}} produces better-styled formula text than not using it, as you can see. Using $...$ for complex formulas is fine. Using {{math}} for simple formulas is also fine. There's a grey area in between where it's less clear. $...$ generally shouldn't be used for simple stuff in running text if it can be avoided, especially for single letters. The Media Viewer also won't render $...$ in image captions, so it should be avoided there if at all possible as well. As I said before, it hinders accessibility and prevents copying to the clipboard. Formulas can be of mixed type in different parts of a page. Simple stuff is often just as easy to type up either way. These comparisons of byte counts are fairly meaningless. –Deacon Vorbis (carbon • videos) 02:43, 18 March 2020 (UTC)
I see that adding {{math}} can make the equations nicer; however, that doesn't address that they're still incredibly hard to type out and therefore are error prone. That's why I brought up bytes and characters; you're pressing the keyboard a hundred times more, and inputting more information for the computer, than you should be for something which looks worse than if you had decided to use LaTeX. Overall, it's time consuming. Now, if someone wants to go ahead and waste their own time typing out poorly formatted math equations, that's fine. But that is at the expense of the interested reader. And another thing: this doesn't address is, again: how are you going to write out an integral? A fraction? LaTeX clearly saves the day. But if you use both math styles, then the issue of mixed math rendering arises which is yet again at the expense of the reader. I can personally say that, years ago when I was first starting out as a math student, I was very confused by the mixed math notation on Wikipedia and was wondering why on earth people were rendering math by italicizing letters of the alphabet. But then I would always be relieved to see LaTeX-ed equations; just like they appear in books, notes, research papers, i.e. literally everywhere a professional mathematician needs to communicate their thoughts through some kind of external medium. Lltrujello (talk) 21:59, 19 March 2020 (UTC)
1. Changing systematically from html to latex is highly time consuming. Althoug the process could be automatized, this is not the case. Moreover, this change requires experimented editors who understand the formulas. Their time would be much more useful, if spent for improving the phrasing of the articles. Surprisingly, this is often the most elementary articles that are badly written (and also that have badly formatted formulas).
2. Changing from raw html (''x'') to {{math}} is fast and easy (select the formula, click on the button {{math}} in the "math and logic" menu, and check special characters like =, |, {, })
3. {{mvar|x}} is easier to type than $x$ (10 vs 14 characters)
4. Inline latex has some alignment issues: ${\displaystyle x}$  for example
Therefore, my practice and my recommendation are
• For new content, use {{math}} for very simple inline formulas, and [itex] otherwise.
• For existing content, add {{math}} to all formulas in raw html that appear in the edited section.
D.Lazard (talk) 10:37, 18 March 2020 (UTC)
Using Visual Studio Code I can copy and paste the existing text and systematically change the math to LaTeX, so it wouldn't be too time consuming. I see and agree with your points, but it is unfortunately more time consuming to swap back and forth between different types of math rendering than just putting everything to LaTeX once in VS Code. I don't want to spend time on this page editing every math equation, deciding whether or not to use {{mvar}} or LaTeX, hence why I proposed a systematic clean up to LaTeX. And LaTeX is pretty and easy to write. But I'll just follow your advice and edit more niche math articles that need more help than just styling updates. Perhaps maybe in the future people will be more open to a full LaTeX switch, but for the now the article will stay a bit ugly, which is unfortunate for the future interested readers. Lltrujello (talk) 21:59, 19 March 2020 (UTC)

## Example for associativity in 2nd Example wrong?

In the example for the associativity in Group_(mathematics)#Second_example:_a_symmetry_group, it seems like the right before left rule hasn't been followed? (fd ∘ fv) gets simplified to r3 while it should probably be r1 if I understood the article correctly. If not then the beginning example explaining the right to left rule is probably wrong. 2A02:908:617:1580:F9FC:1277:BB22:AB42 (talk) 02:30, 16 September 2020 (UTC)

It's right as is. The way the world writes function composition, especially when those functions are group elements, is just plain confusing. Follow vertex 1 for example: ${\displaystyle f_{\mathrm {d} }\circ f_{\mathrm {v} }}$  first applies a vertical flip, so vertex 1 moves to vertex 4, and then a diagonal flip, where vertex 4 stays fixed. So in the end, vertex 1 has moved to vertex 4. Following the other vertices, one finds they're all also rotated by 90 degrees counterclockwise, which corresponds to ${\displaystyle r_{3}}$  in this notation. –Deacon Vorbis (carbon • videos) 02:48, 16 September 2020 (UTC)
Thank you for the quick explanation, I can follow your reasoning (and the examples on the page) looking at the images. I assumed the cayley table also needs to be read right to left, which is where my confusion stemmed from. 2A02:908:617:1580:9CB6:8597:745A:1032 (talk) 03:33, 16 September 2020 (UTC)