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