Groups '93 Galway/St Andrews
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The title of this survey has already appeared in the literature at least twice (see G. Higman [14] and G. E. Wall [35]). As in [14, 35] we will not try to develop some general theory but rather will concentrate on particular group-theoretic problems in which Lie algebra methods proved to be useful.
Our main objects will be finite p-groups and their relations: pro-p groups and residually-p groups.
In §1 we consider residually-p groups whose Lie algebras satisfy polynomial identities. To show that this class is well behaved we sketch the proof that a finitely generated periodic group with this property is finite.
The §2 is dedicated to another “ring theoretic” problem in p-groups: the famous Golod-Shafarevich inequalities.
As we have already mentioned above our main object of interest is a finite p-group. However, since this is too difficult an object to be studied individually, we will study arrays of finite p-groups. More precisely, let G be a group. A system of homomorphisms φi : G → Gi, i ∈ I, is said to approximate G if for any arbitrary element 1 ≠ a ∈ G there exists a homomorphism φi, such that φi(a) ≠ 1. Let Hi = Ker φi. The definition above says that the system of homomorphisms {φi, i ∈ I} approximates G if and only if ∩i∈IHi = (1). In this case we also say that G can be approximated by groups Gi, i ∈ I.
Let p be a prime number. A group G is said to be a residually-p group if it can be approximated by finite p-groups.
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