On the Radicals of a Group that does not have the Independence Property

We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable andwhose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013. The Fitting subgroupof a stable group is nilpotent and definable (F.Wagner [11]). More generally, the Fitting subgroup of a group that satisfies the descending chain condition on centralisers is nilpotent (J. Derakhshan, F. Wagner [5]) and definable (F. Wagner [12, Corollary 2.5], see also [10] and [1]). The soluble radical of a superstable group is soluble and definable (A. Baudish [2]). Whether this also holds for a stable group is still an open question. Inspired by [9], we provide an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable and whose soluble radical is neither soluble nor definable. The proofs require some algebra because we have decided to provide a precise computation of the Fitting subgroup and soluble radical of the group considered. Definition 1 (independence property). LetM be a structure. A formula φ(x, y) has the independence property inM if for all n ∈ , there are tuples a1, . . . , an and (bJ )J⊂{1,...,n} of M such that ( M |= φ(ai , bJ ) ) ⇐⇒ i ∈ J . M does not have the independence property (or is NIP for short) if no formula has the independence property in M . Let L be a first order language, M an L-structure. A set X is interpretable in M if there is a definable subset Y ⊂ M in M and a definable equivalence relation E on X such that X = Y/E. A family {Yi/Ei : i ∈ I } of interpretable sets in M is uniformly interpretable in M if the corresponding families {Yi : i ∈ I } and {Ei : i ∈ I } are uniformly definable in M . Let L be yet another first order language. An L-structure N is interpretable in M if its domain, functions, relations, and constants are interpretable sets in M . Received April 20, 2015.