A mathematical structure is `axiomatizable' if it is completely determined by some family of sentences in a suitable first-order language. This idea has been explored for various kinds of structure, but I will concentrate on groups. There are some general results (not many) about which groups are or are not axiomatizable; recently there has been some interest in the sharper concept of 'finitely axiomatizable' or FA - that is, when only a finite set of sentences (equivalently, a single sentence) is allowed.

While an infinite group cannot be FA, every finite group is so, obviously. A profinite group is kind of in between: it is infinite (indeed, uncountable), but compact as a topological group; and these groups share many properties of finite groups, though sometimes for rather subtle reasons. I will discuss some recent work with Andre Nies and Katrin Tent where we prove that certain kinds of profinite group are FA among profinite groups. The methods involve a little model theory, and quite a lot of group theory.

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.