A pointed-group is an ordered pair (G,C) where G is a group and c is a specific element of G. Thus a pointed-group is a group together with a distinguished element. Pointed- groups may be regarded as a particular type of universal algebras.
The object of the thesis is to generalize ideas and results concerning varieties of groups to varieties of pointed-groups. For example, a law of a pointed-group (G,C) is a word w in the variables y, x1 x2 ‚€¶.such that w takes the value 1 whenever c is substituted for y and arbitrary elements of G are substituted for xl, x2‚€¶.
The principal topic studied is the finite basis question, which asks if the laws of (G,C) are all consequences of some finite set of laws. We prove that the laws of (G,C) are finitely based if G is either nilpotent or metabelian, thereby generalizing theorems of R.C. Lyndon and D.E. Cohn.
However, the main reasons for studying laws of pointed-groups is to provide a test case for a conjecture of Sheila Oates Macdonald which states that if V is any variety of universal algebras whose congruence lattices are modular, then every finite algebra in y has a finite basis for its laws.
One well-known instance is a theorem of Sheila Oates and M.B. Powell, which states that the laws of every finite group are finitely based. It is not difficult to see that the congruence lattice of any pointed-group is modular, therefore, we are led to the study of pointed-groups (G,C) where G is a finite group. In this case we prove that the laws of (G,C) are finitely based, under additional assumptions on G, for example that the Sylow sub groups of G are abelian. However, we have been able to prove that this is not always true. As we have found a finite pointed-group whose set of laws has not a finite basis thereby the generalization of the theorem of Sheila Oates and M.B. Powell to finite pointed-groups is false and thereby providing a counter example to the conjecture of Sheila Oates Macdonald to be false