Consider the set of all bijections from a set to
itself,
*Bij*. If and are two bijections, then so is their
composition . There is the indentity bijection that sends
every element to itself, which we call , and such that if is
any bijection,
. Morever, every bijection
has an inverse, , and the composition of a bijection and its
inverse is equal to the identity.

*** Caesar's code and Borges' library.**
If we consider the set of bijections of the alphabet
,
we enter the world of cryptography. Indeed, Julius Caesar used to
encrypt his messages by substituting the letter by the letter ,
the letter by the letter , and so on (the letter is substituted
by the letter ). The problem with this type of code is that
*ymjd fwj jcywjrjqd jfxd yt gwjfp.* Hint: read
*The Gold-Bug* by Edgar Allan Poe.

In a beautiful short story titled *The Library of Babel,* Jorge
Luis Borges describes a library of books whose pages are filled with
permutations of the letters of the alphabet, commas, points, and blank
spaces. Of course, all the knowledge of the Universe could be found in such
library since even the pages you're reading now are mostly of this form.

In the example of the bijections, we have a set together with an operation (the composition) that takes two elements of the set and gives you another. Moreover, this operation satisfies certain properties. Since this structure comes up very often in mathematics, it is useful to give it a name.

- Multiplication is associative: .
- Existence of an identity: there is an element such that for any , .
- Existence of inverses: for any element , there is an element such that .

- The identity belongs to .
- Whenever an element is in so is its inverse.
- Whenever two elements are in so is their product.

Clearly, if is a subgroup of , then is a group in its own right.