It is an open problem whether the isomorphism class of a finite p-group is determined by its group ring over a field of characteristic p. In this thesis we present the theory of Jennings, which gives invariants of the group ring in terms of the so-called Brauer-Jennings-Zassenhaus series (from now on, just M-series). We then use these invariants to attack modular isomorphism problems.
Defining a suitable function, we give a non-recursive definition of the M-series of an abelian p-group. This yields a new proof that finite abelian p-groups split (that is, are determined by their group ring). We also show that extraspecial p-groups split when p is an odd prime. Applying the so-called "kernel size" technique, we extend a result by Donald Passman which says that D_8 and the quaternions split over GF(2), by showing that these groups do in fact split over any field whose cardinalitu is an odd power of 2. Finally, we split a pair of groups of order p^6.