Automorphisms of κ -existentially closed groups

We investigate the automorphisms of some κ-existentially closed groups. In particular, we prove that Aut(G) is the union of subgroups of level preserving automorphisms and | Aut(G) | = 2 κ whenever κ is inaccessible and G is the unique κ-existentially closed group of cardinality κ. Indeed, the latter result is a byproduct of an argument showing that, for any uncountable κ and any group G that is the limit of regular representation of length κ with countable base, we have | Aut(G) | = ℶκ+1, where ℶ is the beth function. Such groups are also κ-existentially closed if κ is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.
Monatshefte fur Mathematik


Existentially closed groups
Gürel, Yağmur; Kuzucuoğlu, Mahmut; Koçak Benli, Dilber; Department of Mathematics (2021-12-10)
A group G is called an Existentially Closed Group (Algebraically Closed Group) if for every finite system of equations and inequations with coefficients from G which has a solution in an over group H ≥ G, has a solution in G. Existentially closed groups were introduced by W. R. Scott in 1951. The notion of existentially closed groups is close to the notion of algebraically closed fields but there are substantial differences. Existentially closed groups were studied and advanced by B. H. Neumann. In this sur...
Description of Barely Transitive Groups with Soluble Point Stabilizer
Betin, Cansu; Kuzucuoğlu, Mahmut (Informa UK Limited, 2009-6-4)
We describe the barely transitive groups with abelian-by-finite, nilpotent-by-finite and soluble-by-finite point stabilizer. In article [6] Hartley asked whether there is a torsionfree barely transitive group. One consequence of our results is that there is no torsionfree barely transitive group whose point stabilizer is nilpotent. Moreover, we show that if the stabilizer of a point is a permutable subgroup of an infinitely generated barely transitive group G, then G is locally finite.
Automorphism groups of rational elliptic surfaces with section and constant J-map
Karayayla, Tolga (2014-12-01)
In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is a",. The automorphism group of such a surface beta: B -> a"(TM)(1), denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) a < S Aut (sigma) (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut (sigma) (B) of the automorphi...
Beauville structures in finite p-groups
Fernandez-Alcober, Gustavo A.; Gul, Sukran (2017-03-15)
We study the existence of (unmixed) Beauville structures in finite p-groups, where p is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite p-groups satisfying certain conditions which are much weaker than commutativity. This result applies to all known families of p-groups with a good behaviour with respect to powers: regular p-groups, powerful p-groups and more generally potent p-groups, and (generalised) p-central p-groups. In particular, our characterisatio...
Classification of Automorphism Groups of Rational Elliptic Surfaces
Karayayla, Tolga (null; 2011-01-06)
In this paper, we give a classification of (regular) automorphism groups of relatively minimal rational elliptic surfaces with section over the field which have non-constant J-maps. The automorphism group of such a surface B is the semi-direct product of its Mordell–Weil group and the subgroup of the automorphisms preserving the zero section σ of the rational elliptic surface B. The configuration of singular fibers on the surface determines the Mordell–Weil group as has been shown by Oguiso and Shioda (...
Citation Formats
B. Kaya and M. Kuzucuoğlu, “Automorphisms of κ -existentially closed groups,” Monatshefte fur Mathematik, vol. 198, no. 4, pp. 791–804, 2022, Accessed: 00, 2022. [Online]. Available: