On p -groups of low power order

Abstract: We know that groups of order p, wherepis a prime, are cyclic and are all isomorphic to Zp. That there are only two groups of orderp2up to isomorphisms, both of them abelian is also a well known fact. To continue this procedure we will in chapter one classify the structures of p-groups of orderp3andp4. The structure theorem of abelian groups tells us everything about structuring the abelian p-groups as direct products and so we will define another structure operation, namely the semi-direct product, in order to structure the non-abelian ones as well. We conclude that there are five non-isomorphic groups of order p3, three of those being abelian. Furthermore, when p > 3, we find a semi-direct product for all non-isomorphicp-groups of orderp4for which there are 15. Since the semi-direct product uses the automorphism groups of the groups it takes as arguments we will also study the automorphism groups of the p-groups of orderp2andp3. Chapter two will deal with the subgroup structure of the groups discussed in chapter one. We will determine the number of subgroups in each group as well as acquire some knowledge of the relation between the different subgroups. Our approach will be combinatorial, using presentations. The purpose of the final chapter is to study the representations of the non-abelian p-groups of orderp3andp4through their character tables. Methods for obtaining these characters are both lifting characters of abelian subgroups and by use of the orthogonality relations. The conjugacy classes of these groups will be calculated and a short introduction to representation theory will also be given.