We know also that::

1) a cyclic group is simple iff the number of its elements is prime;

2) Abelian simple groups are cyclic;

3) the smallest non-cyclic, but simple, group has order 60.

Greetings, Tadeusz

simple means, there exist no invariant subgroups. A cyclic g is an = e, may be simple. A "small" noncyclic simple group is SU(2), and it contains all g as subgroups.

The remarkable relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical identity (the eigen decom position) that a square matrix A can be decomposed into the very special form

              A = PDP-1                                                                                        (1)

where P is a matrix composed of the eigenvectors of A , D is the diagonal matrix constructed from the corresponding eigenvalues, and P-1  is the matrix inverse of P . According to the eigen decomposition theorem, an initial matrix equation

             AX =Y                                                                                                (2)

can always be written

             PDP-1 X=Y                                                                                        (3)                                                                                                                                                                 

(at least as long as P is a square matrix), and premultiplying both sides by P-1    gives

            DP-1 X=P-1 Y                                                                                     (4)

Since the same linear transformation P-1 is being applied to both X and Y , solving the original system is equivalent to solving the transformed system

            DX' =Y',                                                                                              (5)

where   X'=P-1 X   and  Y' = P-1 Y . This provides a way to canonicalize a system into the simplest possible form, reduce the number of parameters from nxn for an arbitrary matrix to n for a diagonal matrix, and obtain the characteristic properties of the initial matrix. This approach arises frequently in physics and engineering, where the technique is oft used and extremely powerful.

                                                                                           From Wolfram Math Worl

"that a square matrix A can be decomposed into the very special form A = PDP-1  ..." . This, diagonal D, is generally not the case. 

Thank you for all your answers. I did PhD in pure maths. I would like to work on Quantum gravity. Can I apply for another PhD for physics?