Generation of Mathieu groups Mathieu groups are well known (and known long time ago) among the famous sporadic simple groups. However, I find that it is rarely introduced in graduate courses in India, and also student's don't discuss these groups in lecture sessions. One reason could be the availability of material on it with simple lucid language. 
While looking these groups through book of M. Hall, I found it difficult to proceed with his discussion - he says "consider this-this permutation"; but question comes, why this permutation? What is idea behind chosing it? So, I don't find any other source for exposition of these groups.
Can one suggest some elementary exposition of the constructions of the five Methieu groups?
 A: I did some reading on these groups last semester, and I still don't have a great intuition for their internal structure, but I found a couple sources that made me happy.  One was Passman's $\textit{Permutation Groups}$, which has a surprisingly readable proof of the fact that if $G$ acts sharply $k$-transitively on $n$ points with $k \geq 4$, then either $G = S_n$ or $A_n$, or $(k, n) = (4, 11)$ or $(5, 12)$.  (Some steps of the proof tell you about cycle structures of elements, e.g. that involutions in $M_{11}$ must have four transpositions and three fixed points.  A lot of this kind of information is surprisingly easy to deduce from the sharp 4-transitivity alone.)  I think Passman also includes a construction of the larger Mathieu groups that bootstraps upward from $M_{21} = \mathrm{PSL}(3, 4)$.
Very often the best way to understand a group is to understand an object that it acts on, and for this purpose, I found Robin Chapman's survey on Golay codes useful.  Section 1.4 is particularly nice and explicit.
