The (standard) definition of a group.

Question

Edited to incorporate suggestions from the comments and responses:

Typically, the definition of a group is as follows:

Definition: If $S$ is a set, $*$ is a binary operation on $S$, and $e \in S$, then $G = (S,e,*)$ is called a group if

(i) $(ab)c = a(bc)$, $\forall a,b,c \in S$ (associativity);

(ii) $\exists e \in S$ such that $ae = a = ea$, $\forall a \in S$ (identity); and

(iii) $\forall a \in S$, $\exists b \in S$ such that $ab = e = ba$ (inverse).

Consider the following definition.

Definition: If $S$ is a set, $*$ is a binary operation on $S$, and $e \in S$, then $G = (S,e,*)$ is called a group if

(i) $(ab)c = a(bc)$, $\forall a,b,c \in S$ (associativity);

(ii) $\exists e \in S$ such that $ae = a$, $\forall a \in S$ (right identity); and

(iii) $\forall a \in S$, $\exists b \in S$ such that $ab = e$ (right inverse).

It an be shown that these axioms imply that every right inverse is a left inverse and that $e$ is a left identity. (Of course, there's nothing special about using right identity and right inverse and that we could also take left identity and left inverse as axiomatic.)

Question 1: In most undergraduate textbooks in abstract algebra I've seen (I realize this is anecdotal), the first definition is used. Is there a reason that authors use the first definition and not a variant of the second one? This seems strange to me given that it is desirable to make definitions as lean as possible.

Question 2: Alternately, are there textbooks that employ the second definition (or a variant thereof)?

Answer 1

Question 2: Here are four examples from my bookshelves:

1. Derek Robinson's A Course in the Theory of Groups, 2nd Edition (Springer, GTM 80), defines a group as a semigroup (nonempty set with an associative binary operation) that has a right identity and right inverses (page 1; he proves they also work on the left in 1.1.2, on page 2).

2. Marshall Hall, Jr.'s The Theory of Groups (AMS Chelsea Publishing is the version I am looking at). Gives both the two-sided and the one-sided versions as part of the overall definition of a group, indicating the two-sided one is "redundant" and giving the one-sided one.

3. Burnside, Theory of groups of finite order (I'm looking at Dover's 1959 reprint of the 1911 second edition of the original) defines a group asking only that elements have left inverses. He does not require an identity explicitly, but it follows from the requirement for left inverses.

4. van der Waerden's Algebra requires only a left identity and left inverses.

Question 1. If you want to fit groups into the more general theory of semigroups, monoids, etc., then note that in a monoid you must specify that the identity is two-sided; you cannot deduce it. So if you want to say something like "A group is a monoid in which every element has inverses", or "A group is a semigroup that has an identity and also has inverses for every element" (this is basically what Bourbaki does), then your specification of identity must be two-sided, in which case having the inverse be defined one-sided looks a little strange.

Also, the definitions match the experience that most undergraduates will have had: at this point, they are probably familiar with the usual numerical examples ($\mathbb{Z}$, $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}$, maybe the quaternions, possibly the integers modulo $n$), and possibly also matrices. So it's best to give a definition that matches expectations even if it is a bit more involved/superfluous, than one that is formally more inclusive (by putting fewer conditions on the object) but which may seem to invite queries when matched with the usual examples. Especially when one almost immediately would show that they are in fact two-sided.

Answer 2

Technically speaking, neither of your definitions is correct, because (iii) refers to an undefined $e$. From a rigorous point of view, you have 2 options:

Option 1

$(G,*,e)$ is a group iff $*$ is a binary operation on $G$ and $e∈G$ and:

1. $∀a,b,c\ ( \ (a*b)*c = a*(b*c) \ )$.
2. $∀a\ ( \ a*e = e*a = a \ )$.
3. $∀a\ ∃b\ ( \ a*b = b*a = e \ )$.

Option 2

$(G,*)$ is a group iff $*$ is a binary operation on $G$ and:

1. $∀a,b,c\ ( \ (a*b)*c = a*(b*c) \ )$.
2. $∃e\ ( \ ∀a\ ( \ a*e = e*a = a ∧ ∃b\ ( \ a*b = b*a = e \ ) \ )$.

~ ~ ~

That aside, it is wrong to think that it is at all good to have the 'leanest' definition possible. For example, PL (propositional logic) can be axiomatized by a single axiom sentence schema (e.g. Meredith's as given on wikipedia). If you want to make things worse, use the sheffer stroke (NAND) and no other boolean connectives, just because NAND is functionally-complete.

Another example is PA (First-order Peano Arithmetic). The axiomatization of PA in terms of a discrete ordered semi-ring with induction is far superior to the successor-based axiomatization, simply because it reveals more of the true structure of the intended model $ℕ$ of PA. In fact, the motivation for PA in the first place arose from wanting to axiomatize $ℕ$, so we care only about theories that can prove the basic properties of $ℕ$, and unsurprisingly these basic properties are expressed precisely by the discrete ordered semi-ring axioms plus induction.

Answer 3

I cannot answer your second question, but I'll try answering your first.

Even if we were to work with the second definition, you can bet one of the first things any textbook author would do would be to prove that the second definition implies the first and then work with that throughout the rest of the text.

Also, the fact that the two definitions are equivalent is not really seen as important, because it is rarely the case that we start with a structure knowing it satisfies the requirements of the 'right-group' definition but unsure whether or not it is a 'true group'.

In addition, a definition like this is likely to cause more confusion to the reader. While it may be superficially leaner, people find symmetries very intuitive. Defining groups in this way makes them seem strange (even more than they already are to a lot of people) and raises a lot of questions along the lines of 'what if this rule is tweaked?' which are perfectly fine questions to ask, but they detract from what most abstract algebra books are about.

Ultimately, I think if this change in definition was introduced in a book, it could give the reader some interesting questions to think about, but will mostly just take up some extra space towards the start of the book exploring things which will be irrelevant to the rest of the book, after which the author would go on ahead using the standard definition anyway. Which is probably why it isn't used much.

(feel free to comment or edit for any corrections or suggestions)

Answer 4

This is a partial “frame challenge” answer to your Question 1. You write:

This seems strange to me given that it is desirable to make definitions as lean as possible.

“Leanness” is certainly desirable, other things being equal, but it is not the only desirable criterion for definitions, nor the predominant one. Other important ones include naturalness, comprehensibility, and generalisability; and all these make the standard definition of a group preferable. Looking at a few specific issues:

• The standard definition is symmetric. The presentations you suggest introduce an artificial asymmetry in the axiomatisation — of course it still follows that the resulting theory is symmetric, but the axiomatisation hides this.

• In examples of groups, we think of the inverses and units as two-sided, because they are. It’s only natural to split up the left- and right-handed versions in examples where they really do diverge. So the standard presentation better fits how we view examples.

• As Arturo Magidin’s answer details, the standard definition generalises better when we move to weaker structures — monoids, and so on, where the one-sided and two-sided notions really aren’t equivalent.

These are all a bit subjective, but nonetheless very real and important when choosing a definition.

For comparison, Higman and Neumann (following earlier work of Tarski) showed that groups can be axiomatised using a single operation $x/y$ “division”, and a single axiom: $x / ((((x / x) / y) / z) / (((x / x) / x) / z)) = y$. This is certainly “leaner” than either the standard presentation or your single-sided versions; but is also much less comprehensible, natural, or generalisable. I guess you’d agree that this shouldn’t be given as the primary definition of a group. Your suggestion is certainly much better than this one; but I think most mathematicians would agree that for most purposes, it’s slightly less clear and natural than the standard definition, and so the standard one remains preferable.

Answer 5

Deliberately not an answer

I think it's not a bad idea to present the symmetric definition, work with groups a little, and then have an exercise saying "Here's an alternative definition; show that it implies our definition." And that exercise could be followed with a discussion of how/why we choose certain definitions [perhaps cribbed from some of the answers here].

In a course where you're trying to train budding mathematicians, you assign this problem. In a course where you're just trying to teach some algebra (e.g., to folks who need to know it for crystallography), you skip it.

And in a course where you're planning to discuss how to build a hierarchy for the formalization of mathematics and an implementation using computers, like that in Isabelle/HOL, for instance, you can go nuts talking about all the different possibilities, and how each one aids/hinders your development in further aspects of algebra.