Step by step procedure to obtain irreducible representations and  construct character table of a group I am studying group theory and character table of $S_2$ is given in the book.
But how to obtain this table is not given. 
Can someone explain how exactly to construct this table? 
 A: S2 is a group with two elements, call them 1 and t.  A representation of S2 takes 1 to the identity matrix and t to some matrix, so we can think of representations X of S2 as simply some matrix X(t).  Of course not just any matrix will do, since X is a homomorphism and so $(X(t))^2 = X(t^2) = X(1)$ has to be the identity matrix.  In other words, X(t) has to be a matrix whose square is the identity.  All of its eigenvalues also have to square to the identity, so they are all ±1.  By considering the Jordan canonical form of X(t), you can see that X(t) must be diagonalizable (since a large Jordan block does not square to the identity).
Of course if we diagonalize X(t), then each of the standard subspaces is an invariant subspace, so X is only irreducible if it is one dimensional.
However, there are not very many 1×1 matrices whose eigenvalues are just ±1.  They are X1(t) = [1] and X2(t) = [-1].  These are both irreducible representations of S2, so we have found them all.  Writing down their traces we get:
$$\begin{array}{r|rr}
S_2 & 1 & t \\ \hline X_1 & 1 & 1 \\ X_2 & 1 & -1 \end{array}$$
Similar ideas work for any finite abelian group.  One takes a minimal generating set and arbitrarily assigns appropriate roots of unity to the generators, giving one one-dimensional irreducible representation per element of the group.
