Contribution to the theory of abstract groups

Abstract

It is well known that in the theory of abstract groups an important role is played by the free groups of which the quotient groups make up an inexhaustible resource for the abstract groups. There exist, however, some more vast classes of abstract groups which contain the free groups as very particular case; they are the classes of P-groups. Any abstract multiplicative group can, as we know, be given by a set A = {a i }, i ∈ I, of generators and an exhaustive set F of defining relations which connect these generators. Any relation between the elements of A results from the relations F and the trivial relations between the elements of A. There exist some properties named P-properties which can be common to all relations, including the trivial relations connecting the elements of certain sets of generators of multiplicative groups. At present, we know about thirty such properties, to each of them corresponds a vast class of P-groups which yields to an elegant general theory. Here, we present some important classes of P-groups, especially the quasi free groups, the quasi free groups modulo n, the quasi free groups moduli N, the free groups modulo n, the free groups moduli N and the P-symmetric groups. Next, we define the P-products of groups, products presenting some analogies with the free product, and we indicate some properties of these products. Then, we introduce the fundamental and the quasi fundamental groups as well as their bases and we define the rank, the essential invariant, of a fundamental group. To conclude, we review some curious properties of two fundamental groups of rank 2, the first is given by a couple of generators bound by an exhaustive set of two defining relations and the second is given by a couple of generators bound by a single defining relation.