First-Order Recognizability in Finite and Pseudofinite Groups

Abstract

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

Powers of commutators in linear algebraic groups

“…Remark 1.4. J. Wilson gives some other first-order properties of groups that hold for finite groups but not for arbitrary groups [19]; see [18], [5]. One can use the methods of this paper to prove that these properties hold for linear algebraic groups over an algebraically closed or pseudo-finite field.…”

Finite axiomatizability for profinite groups

“…As matrix multiplication in M 2 (U ) is L gp definable in G (4) , it follows that θ is definable. This completes the proof of Claim 2.…”

“…As H and U are definable subsets of G, the mappings u : G → G and h : G → G are both definable. Hence so is θ, because u(1) is the parameter u, u(0 (4) is definable. Obviously.…”

“…Write G = Af 1 (R). We have a visible topological interpretation of G as a subset of M 2 (R) = R (4) .…”

“…For example, a theorem of Felgner shows that this holds for the class of non‐abelian finite simple groups (see [39, Theorem 5.1]), and Wilson [41] shows that the same is true for the class of finite soluble groups. On the other hand, Cornulier and Wilson show in [4] that nilpotency cannot be characterized by a first‐order sentence in the class of finite groups.…”