Eine Herausforderung für die Mathematik(didaktik)?

D. M. Evans, D. Macpherson and A. A. Ivanov: Finite covers;
Z. Chatzidakis: Definable subgroups of algebraic groups over pseudofinite
fields;
W. Hodges: Groups in pseudofinite fields;
D. Lascar: The group of automorphisms of the field of complex numbers
leaving fixed the algebraic numbers is simple;
D. Evans, D. Lascar: The automorphism group of the field of complex numbers
is complete;
P. J. Cameron: The algebra of an age;
M. Boffa: Elimination of inverses in groups;
F. Oger: Model-theoretic properties of polycyclic-by-finite groups;
I. M. Chiswell: Non-standard free groups;
P. H. Pfander: Finitely generated subgroups of the free Z[t]-group on two
generators;
K. Burke, M. Prest: Rings of definable scalars and biendomorphism rings;
B. Kim: Recent results on simple first-order theories.
The main theme is the strong interplay between model theory and group
theory. The first article for instance (72 pages!) starts by explaining how
Zil’ber’s ladder theorem leads to the study of the so called cover problem
which can be described entirely in group theoretic terms. Surprisingly, even
certain cohomology groups can be computed by the careful analysis of this
problem.
From the next two articles let me quote two results after saying that a
pseudo-finite field is one that is a non-principal ultraproduct of finite fields
and that those of characteristic zero represent precisely the infinite models
of the theory of finite fields of characteristic p (p a given prime).
Theorem. Let G be a connected algebraic group defined over F and G 1 a
proper subgroup of finite index of G(F ) which is definable in F . There exists
a connected algebraic group H defined over F and a F -rational surjective
homomorphism g : H → G with finite central non-trivial kernel such that
imH(F ) has finite index in G 1.
Theorem (Matthews, Vaserstein, Weisfeiler). If G is an almost simple, simply
connected closed subgroup of Gl n(Q) and Γ a finitely generated subgroup
of G(Q), which is Zariski dense in G, then but for a finite number of primes
p, Γ/p = G p(F p).
The proof of this theorem is noneffective but much simpler than the
original one.
Lascar’s article contains the proof of the fact that the subgroup of automorphisms
of the complex numbers whose elements fix every algebraic
number is a simple group.
In the next article, using CH, its completeness is proved and generalizations
are given.
Cameron shows how to oligomorphic groups G (more specifically a permutation
group G of an infinite set Ω such that for all n the induced action
on the set of subsets of cardinality n has only finitely many orbits) can be
attached the reduced incidence algebra a la Rota and that it is a polynomial
algebra (in general it is not) for certain such groups.
The next two articles have to do with finite extensions of nilpotent
groups. Indeed, linear groups with elimiation of inverses in identities (i.e.,
one can find a conjunction of monoidal identities) are of that sort.
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