Possible textbook redundancy concerning invertible mappings

Question

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem:

Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity element $\iota_S$. (b) Composition is an associative operation on the set of all invertible mappings in $M(S)$, with identity $\iota_S$.

Here, $M(S)$ denotes the set of all mappings from $S$ to $S$.

My question is simple: doesn't part (a) guarantee part (b)? After all, the set of all invertible mappings in $M(S)$ is simply a subset of all of the mappings in $M(S)$. Nonetheless, the author gives proofs for both parts of this theorem and doesn't address my point above. Where am I going wrong? Could someone confirm my reasoning or show my error by providing a counterexample perhaps?

Answer 1

First things first: It's easier for us if you would cite the result by its number in the book, rather than just saying that it comes from the book. This way, those of us who have the book in hardcopy can easily find it. It is Theorem 4.1 in Chapter I.

Also, the requirement that $S$ be nonempty is useless. I never understood why some people find it helpful to add such requirements. Almost no experts do this; the disease seems to be mostly spread across undergraduate-level texts.

Now, as for your question: You're saying that the associativity of the composition on invertible mappings follows from the associativity of the composition on all mappings, and thus its statement is redundant. This is true, and Durbin himself argues this way in the last paragraph of this proof. However, the statement that "Composition is an associative operation on a set $X$" doesn't just mean that composition is associative. It also means that composition is a welldefined binary operation $X \times X \to X$ (at least in this context); that is, it says that the composition of two elements of $X$ is again in $X$. So, in (b), it says that composition of two invertible mappings from $S$ to $S$ is again an invertible mapping from $S$ to $S$. This is what is really the point of part (b).

Answer 2

Once associativity is established for the larger set, it follows immediately for the smaller set too. The things to worry about here are that composition of invertible mappings is again invertible (so that indeed the operation of composition restricts to the smaller subset) and that the identity mapping is invertible (so that it is present in the smaller subset). These facts indeed guarantee (trivially) that the smaller set has the stated properties. There is no need to re-do all the proof.

Answer 3

The key word here is "operation", not "associative". The associativity of the subset is, indeed, obvious. The fact that it is a binary operation on $M(S)$ is NOT so, and requires demonstration: that is if we call the invertible set of mappings $S \to S,\ A(S)$, that $f,g \in A(S)$ then implies $f \circ g \in A(S)$.

The phrase "...with identity $\iota_S$", seemingly tacked on as an after-thought, is also one of the sticking points, it is not clear that $\iota_S$ is even in the smaller set $A(S)$.

Durbin appears to be hampered at this point by a lack of available terminology: what he is saying is that $M(S)$ forms a monoid, and that $A(S)$ is a sub-monoid of $M(S)$, which has the further structure of a group. That is:

For any (non-empty) set, $S$, the set $M(S)$ forms a monoid, called the monoid of transformations of $S$ (the requirement that $S$ be non-empty is mostly to avoid the ontological murkiness of what a function $f: \emptyset \to \emptyset$ "is". I assure you that you aren't missing anything exciting, there).

Similarly, the the set $A(S)$ forms a group called the group of permutations of $S$. These two structures are fairly important throughout mathematics, especially for finite sets. If $|S| = n$, then $|M(S)| = n^n$, and $|A(S)| = n!$, and these "sizes" are used in various ways to determine things like: "how many ways can I do (something or another)?".

These two structures can be thought of as the "blueprint" for the structure of monoids, and groups, in general, in much the same way as $\Bbb R^n$ is the blueprint for any $k$-dimensional real vector space, where $k \leq n$. They are very important, although you may be absorbed by other particulars of abstract algebra for a while.

In abstract algebra in general, given a "thingy" $A$, with a subset $B$ of $A$, some features of $A$ are "inherited" by $B$, and some are not. The ones that are not, are typically "closure" properties, or the inclusion in $B$ of certain "distinguished elements" of $A$.