Talk:Group (mathematics)

Closure

Although it is important to mention closure, there are a few things that disturb me about the way the definition of group is currently written. What is an operation, before the closure axiom is imposed? A function from G × G to some unspecified set? Not only is this a little vague, but it also contradicts the binary operation page it links to. Also, technically speaking, the sentence defining group is wrong, because it ends before any of the axioms are imposed.

I would propose the following, which is slightly longer, but more explicit about the role of closure, which really should be separate from the group axioms. This also breaks the definition into more manageable chunks: first understand what a binary operation is, and then understand the definition of group. Also, this would bring this page more in line with other Wikipedia pages, such as ring. Finally, there are many modern textbooks at all levels that present the definition along these lines (e.g., Artin, Lang, ...); I would add such references.

A binary operation ⋅ on a set G is a rule for combining any pair a, b of elements of G to form another element of G, denoted a ⋅ b. (The property "for all a, b in G, the value a ⋅ b belongs to the same set G" is called closure; it must be checked if it is not known initially.)

A group is a set G equipped with a binary operation ⋅ satisfying the following three additional requirements, known as the group axioms:

Associativity

For all a, b, c in G, one has (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Identity element

There exists an element e in G such that, for every a in G, the equations e ⋅ a = a and a ⋅ e = a hold. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. For each a, the b is unique (see below) and it is commonly denoted a⁻¹.

^ b: Formally, a binary operation on G is a function G × G → G.

I would welcome advice about which defined terms should be bold and which should be italicized; I'm not sure what the convention is.

Ebony Jackson (talk) 02:49, 16 December 2020 (UTC)

I essentially agree, and I have edited the article accordingly. By the way, I have copy-edited the whole section for clarification and for using a simpler wording that is also more common in mathematics.

About "closure": the term is normally used for the restriction of a binary operation to a subset. Using it as it was done is thus an error. I guess that editors were confused by the usual definition of a subgroup as a nonempty subset on which the group operation and the inverse operation are closed. Using this definition, it is a theorem that a subgroup is a group, and that the groups axioms are thus satisfied. D.Lazard (talk) 10:44, 16 December 2020 (UTC)

The above wording of the last two axioms combines an axiom (one sentence) with consequent properties (e.g. uniqueness of the identity element) that is not part of the axiom. It would be good if this separation was made clearer to the reader, since the current presentation does not adequately distinguish for the reader who is not already familiar with the exact axioms. The parts that do not form part of the axiom could be moved to under the listed axioms, for example, or preceded by "This implies that ...". —Quondum 11:33, 4 May 2021 (UTC)

I'm very tempted to add Closure as one of the four group axioms, as it's already one of the "abelian group axioms". Technically the only difference is the commutativity of the operation, so it doesn't make sense to list closure as an axiom of one but not another. IBugOne (talk) 14:20, 29 December 2021 (UTC)

Please don't: "closure" is a property of subsets, and there is no subset here. The fact that the result of the operation belongs to the group is a part of the definition of an operation. By the way, I have removed the use of "closure" in abelian group#Definition. D.Lazard (talk) 15:00, 29 December 2021 (UTC)

Thank you, IBugOne, for pointing out the discrepancy. I agree with D.Lazard that the best solution to the issue you raise is that closure should not be listed an axiom either for group or for abelian group. Ebony Jackson (talk) 23:01, 29 December 2021 (UTC)

But the set G is a subset of some greater set(s). It is ridiculous to limit closure to subsets, somehow, as if that were even possible? Besides, closure of an operation is exactly the same property whether done on sets or subsets.

Hmm, the closure article gives the example of the set of natural numbers under addition and subtraction - one is closed, one isn't - but no subsets involved.

What is more ridiculous is to have an elementary article on groups which doesn't even mention closure, but to fob it off as part of "binary operation".

Generally, a "binary operation" implies that for two input sets A and B, for all a,b in A,B a*b is defined. That's it. The results don't have to be in either input set.

A "binary operation on a set" implies a single input set S s.t. that for all a,b in S, a*b is defined. Strictly, no more, but often closure may be understood _if that understanding is pointed out or defined locally_.

However, closure is not a _necessary_ part of a "binary operation on a set" - or what would you call a binary operation which is not closed but where the input sets for the two inputs are the same set?

You can call the group operation a "closed binary operation on a set" if you like (I don't).. but I think you must at least mention closure.

Downloaded 5 pdfs on elementary group theory at random to see. Two use the word closure as an axiom, one uses closure as an axiom but without the word, two wrap closure in with "binary operation" or "composition" by defining those phrases locally as requiring closure.

So, is closure an axiom of a group? It's definitely a required property of the group operation. Maybe it's so simple and obvious that people forget to mention it? 62.3.121.230 (talk) 04:33, 9 June 2025 (UTC)

Including both left and right identity and inverse is very common mistake. The existence of the left identity and inverse can be proven using the right identity and inverse and vice versa. So it is sufficient to present only one of each in the list of the axioms. Here there are some proves, for example: https://math.stackexchange.com/questions/65239/right-identity-and-right-inverse-in-a-semigroup-imply-it-is-a-group Andrewsk (talk) 00:06, 20 January 2023 (UTC)

You are right that some of the axioms could be deduced from the others, but this is not a "mistake". The standard textbooks intentionally require the identity be a two-sided identity and so on, presumably because it is more natural not to favor one side. Therefore we should leave it as is. Ebony Jackson (talk) 00:03, 23 January 2023 (UTC)

In a similar vein, I modified the leading sentence to mention that the binary operation is closed (defined on the set). Seeing as the original sentence didn't call it a "binary operation" and instead called it an "operation that combines any two elements to form a third element", I would argue that in order to make this expansion clear and precise, it's required to mention that the domains/codomain are all in the set. So therefore I modified it to "an operation that combines any two elements of the set to produce a third element of the set". Quohx (talk) 06:58, 14 March 2022 (UTC)

Identity and also inverse elements must be part of set

Note to 100.36.106.199 who removed (2 days ago) my words "the set contains an identity element" and returned to the previous wording "an identity element exists": The point is that it is not sufficient for an identity element to exist; it must be part of the set or else the set does not constitute a group.

Consider the first example: the integers under addition. If we consider the set without the identity element zero: ..., -3, -2, -1, +1, +2, +3, +4, ... then we have a set which is NOT a group. Zero still exists but it has to be included in the group.

As for requiring parallelism in wording for identity element and inverse elements, I actually agree that the wording should be parallel. So I will now make it parallel by adding that the inverse elements also must be part of the group (although you said you hoped not). Again for the integers under addition: the set 0, +1, +2, +3, +4, ... is NOT a group without the negative integers. The fact that they exist is not sufficient. Dirac66 (talk) 02:01, 10 July 2023 (UTC)

Your example is incoherent: the object you have presented is not a set with an operation on it (because what is -1 + 1?). Assuming you had not made this error, you would be wrong that an identity exists: the operation is defined (only) on the set, things outside the set cannot be combined using the operation with things in the set and so in particular they cannot be an identity or an inverse. --100.36.106.199 (talk) 13:49, 10 July 2023 (UTC)

I mean, it is true that students first learning abstract algebra suffer from the confusion that you are expressing here. But I think it is instructive that the first time you made the change, you did not even notice that the same argument applies to inverses as to the identity. That's because the meaning is not actually ambiguous or otherwise problematic. --100.36.106.199 (talk) 13:52, 10 July 2023 (UTC)

Having said all that: the revised wording seems fine. --100.36.106.199 (talk) 13:54, 10 July 2023 (UTC)

Undefined terms and notational elements

The statements about injective homomorphisms use several notational elements that have not been introduced previously and that will not be intuitive to a general reader: ${\stackrel {\sim }{\to }}$, $\hookrightarrow$, and $\ker \phi$.

The latter also appears in the Presentations section, along with reference to the free group Xphileprof (talk) 22:39, 20 July 2023 (UTC)

What is the fundamental group of a plane minus a point?

"The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers." I know very little about groups that I didn't learn from this page... but... the integers are a set, not a group, right? So "isomorphic to the integers" is a vague way of saying "isomorphic to some group that has the integers as the underyling set"? — Preceding unsigned comment added by 2404:4408:6A6E:7000:E48B:A59:8E82:2FCF (talk) 08:21, 3 October 2023 (UTC)

In this case, the relevant group is the integers under the addition operation. –jacobolus (t) 17:24, 3 October 2023 (UTC)

The given quote could be considered incorrect: the loop space consists of loops and the fundamental group crucially consists of equivalence classes of loops. The main text avoids this by saying that "elements of the fundamental group are represented by loops" which is perfectly correct but maybe overly evasive or obscure for most readers. Also, the loops don't have to go around the missing point - they just have to avoid it.

There's also the problem that the blue and orange curves in the image don't show two elements of the fundamental group: they show two different (free homotopy classes of) maps from the circle into the space. A loop representing the fundamental group has (although usually only implicitly) a fixed base point, and these two loops obviously have no common base point. So the picture is not quite illustrative of the fundamental group, even though any reader already familiar with the concepts can easily see what it's trying to communicate.

Being fully precise would obviously not be desirable in the context of the page, but perhaps a talented writer could find a way to rephrase the paragraph and image/image caption in a way that remains concise and readable but is also fully accurate. (I'm not talented enough.) Maybe it would help to move the paragraph to its own subsection "algebraic topology" or "fundamental group". Gumshoe2 (talk) 19:31, 3 October 2023 (UTC)

Looking at this picture, I agree it's weird. We probably instead want something like the pictures in Winding number § Intuitive description. –jacobolus (t) 19:37, 3 October 2023 (UTC)

@Gumshoe2 I tried rewriting the explanation here. Is that any clearer? –jacobolus (t) 22:13, 3 October 2023 (UTC)

One can show that

@D.Lazard: I removed the bold text from "...assuming associativity and the existence of a left identity [...] and a left inverse [...] for each element [...], one can show that every left inverse is also a right inverse of the same element as follows.", which you reverted with the comment "It must be clear that a proof is behind the assetrion". I do not understand the need for including "one can show that". Of course it has been shown. That is the reason we know it is true. Is there a way it could be true without having been shown to be true? Nuretok (talk) 13:20, 26 May 2024 (UTC)

This is true, but it is not an evidence. See your talk page. D.Lazard (talk) 13:26, 26 May 2024 (UTC)

Error in examples: division over reals has quasi-group structure despite no closure.

In the final section of the article (Generalizations) there is a table on operations over different sets. Division under the reals is listed as closed but the group structure is given as "quasi-group" despite the table above clearly stating that closure is necessary for a quasi-group structure. Perhaps it could be made more clear what is meant by the table.

Edit: After looking over the table a bit more there are a few confusing things about it. It is not explained why something is "N/A". I found myself double checking many things as I am not a group theorist. Nathalene (talk) 21:11, 24 October 2024 (UTC)

Best definition for lead?

Some time recently, the first sentence of the article was edited to inline the definition of "binary operation" into the definition of "group", presumably to make it more accessible. I'm not sure what to do about the phrase "as does every binary operation" though.

1. If it's removed, more experienced mathematicians reading the page might be misled by the redundancy. That is, how is a group's operation different from a binary operation? (It isn't.)
2. As is, it feels like too much detail for the first sentence, but if it's removed, the sentence reads like a run-on.
3. I don't want to revert the change altogether as it's been in place for a while and I understand the rationale.

(Added to talk page per @D.Lazard's comment) A mentally disabled mathematician (talk) 16:09, 9 February 2025 (UTC)

I agree with your concerns. IMO, a readeer who is confused with the use without definition of "operation" would certainly be also confused with "associative", "identity element" and "inverse". Also, the first line of Binary operation provides an easily accessible definition. So, I have removed the parenthetical comment, and tthe definition of a binary operation. D.Lazard (talk) 16:54, 9 February 2025 (UTC)

Lead has become disjointed

Comparing the current lead section to the version from 2021 that passed FAR, I find that it is disjointed, with several short paragraphs whose meanings have drifted and that do not fit together well. I would like to tentatively suggest rolling it back to this earlier version (and then if anyone feels there are clear improvements that have been made, re-implementing those starting from the older base). What do you think? --JBL (talk) 21:10, 9 June 2025 (UTC)

Comment: Who is the target audience for this article? I think (I may be wrong, though) that both the reverted version lead and the current version lead are way too hard for most high-school students to understand, even those who like math a lot. It would take a lot of effort for them to understand even the first paragraph. But I don't know how to solve this. Undergraduate students would be able to understand with the help from lectures from university... I don't remember exactly where I've read in Reddit someday, but (as I remember) the Soviet mathematician V. I. Arnol'd used to advocate for presenting the group concept as a group of transformations (a group whose elements are maps from some set to itself or something like that). But I don't have a source for that. It would be cool if we could find some way to discover if high school students can grasp the lead (which looks perfectly clear and fine to a professional mathematician or professor, but I think most students couldn't understand very easily). Anyway, this is a great article, I'm not saying otherwise, it's really I don't know how to solve the problem that math like this is impenetrable for most people (more than 99% of the population). Best, Esevoke (talk) 22:09, 9 June 2025 (UTC)

Anyway, your preferred version is better, I think. It's more clear. Esevoke (talk) 22:10, 9 June 2025 (UTC)

I've found the opinion of Arnold: search for "What is a group? Algebraists teach ...". Cheers, Esevoke (talk) 22:14, 9 June 2025 (UTC)

P.S. I think the section "Definition and illustration" is amazing. Btw, a page citing a lot of nice references with different points of views: https://mathoverflow.net/questions/141488/what-books-approach-group-theory-through-transformation-permutation-groups . Esevoke (talk) 23:40, 9 June 2025 (UTC)

I'll try to write my thoughts about the current lead later. I am very tired at this moment, but in a few hours I'll write a better response! Best, Esevoke (talk) 00:19, 10 June 2025 (UTC)

I'll answer tomorrow, I am now too tired and need to sleep. Cya! Esevoke (talk) 02:31, 10 June 2025 (UTC)

Btw, I don't think that generating sets should be in the lead at all (much less in the second paragraph). We should talk more about history and importance or another thing in the lead. In the shoes of the reader: they don't yet grasped what a group is and we already are talking about generating set of a group. Esevoke (talk) 08:04, 10 June 2025 (UTC)

I can't prove it, but the lead is too hard the way is is now. Like I said below: "I don't know how to solve this", haha. But I'll try a little modification. Esevoke (talk) 08:08, 10 June 2025 (UTC)

I'll quote Pier Luigi Ferrari (Ferrari, P. L. (2003). "Abstraction in mathematics". Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 358(1435), pp. 1225–1230. https://doi.org/10.1098/rstb.2003.1316 ): "The example of group theory is one of the most frequently displayed and is appropriate to show the power of abstraction." I think that something more or less by these lines should be empathized in the lead. I don't know, sorry. Esevoke (talk) 08:21, 10 June 2025 (UTC)

I've found a (way more basic) article from Brilliant (website) about groups here a link. I think that their choice of words for the definition is more natural and digestible than ours. Esevoke (talk) 10:53, 10 June 2025 (UTC)

I've changed the lead's beginning, I don't know if all people will agree the change is an improvement (I think it is, 15 years old layman can understand it better this way, I think). Well, I'm done editing this article. Thanks. Best, Esevoke (talk) 12:02, 10 June 2025 (UTC)

The parentheses should be removed from the first few sentences. The current version is choppy and hard to read. –jacobolus (t) 16:16, 10 June 2025 (UTC)

I'll remove them. (still will need more improvements, but the previous one was very difficult for non-mathematicians.. we have to use more common words, and make analogies before presenting a rigorous definition I think) Esevoke (talk) 17:40, 10 June 2025 (UTC)

I won't insist on my version of the first paragraph, feel free to revert to the old version. It was nice to try though. Esevoke (talk) 17:47, 10 June 2025 (UTC)

Alternative definitions of "group"

https://planetmath.org/alternativedefinitionofgroup

Maybe there are more that could be mentioned in the article.. I don't know. Just an idea. Esevoke (talk) 17:50, 10 June 2025 (UTC)

Of course, they should be all equivalent. Esevoke (talk) 17:51, 10 June 2025 (UTC)

Definition

I can't remember if I mentioned this before, but one thing I don't like about the current wording of the definition of group is that it makes it seem as if the uniqueness of $e$ and inverse elements is part of the axioms. I think that the uniqueness is usually a consequence of the axioms, not an axiom itself. May I propose the following revision of the axioms?

Associativity

For all ⁠$a$⁠, ⁠$b$⁠, ⁠$c$⁠ in ⁠$G$⁠, one has ⁠$(a\cdot b)\cdot c=a\cdot (b\cdot c)$⁠.

Identity element

There exists an element $e$ in $G$ such that, for every $a$ in ⁠$G$⁠, one has ⁠$e\cdot a=a$⁠ and ⁠$a\cdot e=a$⁠, and such that the following axiom holds:

Inverse elements

For each $a$ in ⁠$G$⁠, there exists an element $b$ in $G$ such that $a\cdot b=e$ and ⁠$b\cdot a=e$⁠.

Comments: The element $e$ is unique (see below); $e$ is called the identity element (or sometimes neutral element) of the group. For each ⁠$a$⁠, the $b$ in the last axiom is unique (see below); $b$ is called the inverse of $a$ and is commonly denoted ⁠$a^{-1}$⁠. Ebony Jackson (talk) 02:19, 29 December 2025 (UTC)

Normally when one writes axioms they can be considered as separate mathematical statements. But here the "Inverse elements" axiom is nested within the existential quantifier of the "Identity element" axiom. So you can't really format these claims as three separate axioms the way you have. Really you have only two axioms, associativity and a messy complicated identity+inverse thing. I prefer the simpler standard axiomatization. —David Eppstein (talk) 08:23, 29 December 2025 (UTC)

Yes, that's true. Do you have another suggestion for making clear that the uniqueness is not part of the axioms? Maybe we could keep the axioms as they are as you suggest, but just put an explanatory sentence after the axioms saying "The uniqueness of the identity and inverse elements is not part of the axioms; it is a consequence (see below)."

In fact, I think the better definition of group is the one in which the multiplication, identity, and inverse map are all part of the data, and the axioms involve all of those. But this is not usually how it is done in introductory textbooks, so it is probably better not to do that here. —Ebony Jackson (talk) 19:00, 30 December 2025 (UTC)

This makes sense, though I've never liked the "usual" axiomatization of groups, generally preferring the weakest axioms. One really only needs associativity, the two-sided quasigroup axiom (unique existence of a left and right multiplier for every element to get any other), and non-emptiness. One does not need to axiomatize existence of an identity or a two-sided inverse. But this is so unusual that many would not recognize it, so one is left falling back to giving familiar axioms rather than purist axioms. —Quondum 23:45, 31 December 2025 (UTC)

OK, I added the note about uniqueness not being part of the axioms. Yes, many other axiomatizations are possible, but I agree with you, Quondam, that we should stick with the familiar ones. —Ebony Jackson (talk) 00:17, 2 January 2026 (UTC)

IMO, it works nicely with that clarity. But: would it not be better to put all the consequences of the axioms (and hence the names "identity element" and "inverse element") after the axioms with your note, instead of woven in with them? So:

Associativity

For all ⁠$a$⁠, ⁠$b$⁠, ⁠$c$⁠ in ⁠$G$⁠, one has ⁠$(a\cdot b)\cdot c=a\cdot (b\cdot c)$⁠.

Identity

There exists an element ⁠$e$⁠ in ⁠$G$⁠ such that, for every ⁠$a$⁠ in ⁠$G$⁠, one has ⁠$e\cdot a=a$⁠ and ⁠$a\cdot e=a$⁠.

Inverse

For each ⁠$a$⁠ in ⁠$G$⁠, there exists an element ⁠$b$⁠ in ⁠$G$⁠ such that ⁠$a\cdot b=e$⁠ and ⁠$b\cdot a=e$⁠.

Uniqueness of the identity and uniqueness of inverse elements required by any single axiom; they are consequences of the three axioms:

• An element ⁠$e$⁠ that satisfies the identity axiom is unique in ⁠$G$⁠. It is called the identity element (or sometimes neutral element) of the group.
• For each ⁠$a$⁠, an element ⁠$b$⁠ that is an inverse of ⁠$a$⁠ is unique; it is called the inverse of ⁠$a$⁠ and is commonly denoted ⁠$a^{-1}$⁠.

—Quondum 16:24, 2 January 2026 (UTC)

Hmm, not quite. The axiom of inverses does not work, as it implicitly requires a unique identity (or: its use of an identity is ill-defined). These axioms are therefore not quite right, even as they are currently in the article. Definitely needs thought. —Quondum 16:27, 2 January 2026 (UTC)

Yes, what you noticed is the reason for keeping the mention of the uniqueness in the second axiom. Maybe I can try to add something that will satisfy you. —Ebony Jackson (talk) 21:47, 2 January 2026 (UTC)

The alternative definition is something like this, which is logically clearer, but maybe not as common:

Alternative definition

A group is a set $G$ together with

• a binary operation on ⁠$G$⁠, here denoted "⁠$\cdot$⁠", that combines any two elements $a$ and $b$ of $G$ to form an element of ⁠$G$⁠, denoted ⁠$a\cdot b$⁠,
• an element $e$ of $G$, and
• a map $i\colon G\to G$ sending each element $a$ of $G$ to an element $a^{-1}$ of $G$,

such that the following three requirements, known as group axioms, are satisfied:

Associativity

For all ⁠$a$⁠, ⁠$b$⁠, ⁠$c$⁠ in ⁠$G$⁠, one has ⁠$(a\cdot b)\cdot c=a\cdot (b\cdot c)$⁠.

Identity

For all $a$ in ⁠$G$⁠, one has ⁠$e\cdot a=a$⁠ and ⁠$a\cdot e=a$⁠.

Inverse

For all $a$ in ⁠$G$⁠, one has $a\cdot a^{-1}=e$ and ⁠$a^{-1}\cdot a=e$⁠.

The element $e$ is called the identity element (or sometimes neutral element), and $a^{-1}$ is called the inverse of $a$.

Note: Given $G$ and an associative binary operation ⁠$\cdot$⁠, there is at most one element $e$ satisfying the identity axiom, and then at most one map $i$ satisfying the inverse axiom; see uniqueness of the identity and uniqueness of inverses below. Thus an equivalent definition would be a set $G$ with a binary operation ⁠$\cdot$⁠ such that there exists an element $e$ and map $i$ making the three axioms hold. —Ebony Jackson (talk) 22:36, 2 January 2026 (UTC)

References

FWIW, this is the one that's closest to my mental model of a group, in part because the use of only universally-quantified equational laws conforms with universal algebra. But much more important is whether it's covered that way in standard sources. —David Eppstein (talk) 22:45, 2 January 2026 (UTC)

I think that it may be more intuitive and understandable to say (if one follows the axiomatic style):

• For all ⁠$a$⁠, ⁠$b$⁠, ⁠$c$⁠ in ⁠$G$⁠, one has ⁠$(a\cdot b)\cdot c=a\cdot (b\cdot c)$⁠.
• There exists an element ⁠$e$⁠ in ⁠$G$⁠ such that
  • for every ⁠$a$⁠ in ⁠$G$⁠, ⁠$e\cdot a=a$⁠ and ⁠$a\cdot e=a$⁠, and
  • for every ⁠$a$⁠ in ⁠$G$⁠ there exists an element ⁠$b$⁠ in ⁠$G$⁠ such that ⁠$a\cdot b=e$⁠ and ⁠$b\cdot a=e$⁠.

There is no need to make the axioms separate. This has two axioms, the second using a conjunction. I also like the universal algebra approach, but this might be getting away from the simplicity and familiarity that many readers will understand most easily, as the signature of a group is then unfamiliar. —Quondum 23:01, 2 January 2026 (UTC)

This is more or less what I had suggested at the top of this section, with the nesting written out more explicitly. But I think David Eppstein is right that standard sources usually don't spell things out this way, even if it is more logically precise. Maybe it's best just to leave the definition in the article as it is. —Ebony Jackson (talk) 23:29, 2 January 2026 (UTC)

Don't you think that this is exactly what the sources say, albeit with the "sameness" of ⁠$e$⁠ in the last two "axioms" implied, rather than made explicit as I have done? After all, we do not insist on word-for-word copying of the sources, and this is a very direct paraphrase in way that is less taxing on the reader's thinking than the sources are. —Quondum 23:34, 2 January 2026 (UTC)

Some references (Lang Algebra, Bourbaki) define monoid first and prove that the identity is unique before defining group. Other references (Hungerford Algebra, Milne's Group theory notes) write the axioms more or less as we have them, with the implicit presumption that the e in the third axiom is an identity element. I haven't seen a source that explicitly writes out a nested axiom. —Ebony Jackson (talk) 00:54, 3 January 2026 (UTC)

If you're comfortable with that, I'll leave it at that. —Quondum 13:11, 3 January 2026 (UTC)

I am a retired mathematician who has taught Group Theory for many years. You all seem to be doing a fine job, but if you ever want any help, I'll be happy to volunteer. Rick Norwood (talk) 15:34, 3 January 2026 (UTC)

Rick Norwood, simply engage in discussion here and edit the article as you see fit and feel inclined. Edits to this article should mostly obtain consensus on this talk page, since it is a featured article, but see WP:BOLD. —Quondum 22:00, 15 January 2026 (UTC)

Thanks.Rick Norwood (talk) 20:03, 16 January 2026 (UTC)

Jacobson's definition

I cannot access Artin (2018) or Lang (2005), but Lang (2002) does indeed define it badly: that defines a monoid without a labelled identity element, then proves that there is only one identity element, and then defines inverses with respect to e without defining e. If one writes this via formal quantifiers, the e in the definition and proof are bound variables, and hence cannot be referenced outside those quantified statements. To allow this, one could add another axiom (or definition) that effectively says that, if (and since) the identity element is unique, we will label it e (unbound), but this is omitted by Lang.

The reason that I am reopening this is that I have found a source with a clear definition that very closely parallels the others, but does not suffer from this shortcoming: Jacobson, Nathan (1985), Basic Algebra I (2nd ed.), W. H. Freeman (and Dover), ISBN 978-0-486-47189-1. His approach is to define a monoid in a manner that labels e in an unbound way, leaving it usable in the third axiom without a logical reference failure. Summarizing his definitions of monoid and group:

A group is a set G, with a binary operation ⋅ on G that combines any two elements a and b to form an element a ⋅ b, and an element e of G such that:

Associativity

For every a, b, c in G, one has (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Identity

For all a in G, one has e ⋅ a = a and a ⋅ e = a.

Invertibility

For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e.

Note that this hangs together cleanly and does not even need the interspersed comments: these can follow the definition as observations. This is achieved by dint of changing the prelude of the axioms with "and an element e of G" that provides an (logically unbound) label. Jacobson does this in a slightly more formal way by referring to a triple: "A monoid is a triple (M, ⋅, e) ..." and "A group G (or G, ⋅, e) is a monoid all of whose elements are invertible."

Any objections to me making this change? —Quondum 21:45, 14 January 2026 (UTC)

That definition (specifying G, ⋅, and e) is halfway between the current definition (specifying only G and ⋅), and the "alternative definition" above (specifying G, ⋅, e, and the inverse map). There is nothing wrong with that definition, though I think it is true that most authors use something closer to the current definition with G and ⋅ only. —Ebony Jackson (talk) 20:42, 16 January 2026 (UTC)

Agreed, it is very similar, and I think it is logically coherent too. The inverse map is perhaps less standard (except in universal algebra), and and I assumed you'd not pursued that from lack of finding reputable "standard" sources. I do not agree with "more common" as a criterion as long as it is "standard" (or in David's words: "more important is whether it's covered that way in standard sources"), and I think I've covered that base with the reference. We have some liberty to select between good sources, especially if there is a clear reason. Here I believe there is a clear reason, which I gave above, and has the same flavour as one of David Eppstein's objections relating to quantification: the statement as it is given by the article and Lang is literally malformatted from a logical perspective concerning quantification. —Quondum 21:27, 16 January 2026 (UTC)

I have no objection to using this definition from Jacobson. It appears valid (with respect to the nested quantifer issue that causes problems with some other definitions) and is properly sourced. It is maybe not a minimal definition of a group but we cover the idea of defining groups with fewer assumptions in the "Equivalent definition with relaxed axioms" section and in any case for a starting point a more natural definition is a better choice than an unnaturally cut-down definition. —David Eppstein (talk) 21:35, 16 January 2026 (UTC)

The criticism of Lang (2002) is not justified, since Lang is explicit near the bottom of page 3 that in this chapter he is going to use e to denote the unit element. I do not see anything wrong with Lang's definitions (unless one wants to get really pedantic), so they should not be used as a straw man to favor Jacobson's definition. That being said, I do not object to using either the definition we currently have or Jacobson's. I'd suggest waiting a day or two to let other editors voice their opinions, before acting. —Ebony Jackson (talk) 22:17, 16 January 2026 (UTC)