Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F-projective resolution for Z when F is the family of all subgroups HG with rkHrkG-1. We answer this question negatively by calculating the relative group cohomology FH*(G, ?(2)) where G = Z/2xZ/2 and F is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology FH*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family F. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E-2 page is isomorphic to the relative group cohomology of G.


Locally finite groups and their subgroups with small centralizers
ERSOY, KIVANÇ; Kuzucuoğlu, Mahmut; Shunwatsky, Pavel (2017-07-01)
Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of rank at least 3 such that C-G(A) is Chernikov and C-G(a) involves no infinite simple groups for any a is an element of A(#). We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of PSLp(k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that C-G(A) is Chernikov and C-G(A) Keywords: involves no inf...
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...
On local finiteness of periodic residually finite groups
Kuzucouoglu, M; Shumyatsky, P (2002-10-01)
Let G be a periodic residually finite group containing a nilpotent subgroup A such that C-G (A) is finite. We show that if [A, A(g)] is finite for any g is an element of G, then G is locally finite.
Value sets of folding polynomials over finite fields
Küçüksakallı, Ömer (2019-01-01)
Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Beauville structures in p-groups
Gül, Şükran; Ercan, Gülin; Fernández-Alcober, Gustavo Adolfo; Department of Mathematics (2016)
Given a finite group G and two elements x, y in G, we denote by Sigma(x,y) the union of all conjugates of the cyclic subgroups generated by x, y and xy. Then G is called a Beauville group of unmixed type if the following conditions hold: (i) G is a 2-generator group. (ii) G has two generating sets {x1,y1} and {x2, y2} such that Sigma (x1, y1) intersection Sigma(x2, y2) is 1. In this case, {x1, y1} and {x2, y2} are said to form a Beauville structure for G. The main purpose of this thesis is to extend the kn...
Citation Formats
S. Pamuk, “RELATIVE GROUP COHOMOLOGY AND THE ORBIT CATEGORY,” COMMUNICATIONS IN ALGEBRA, pp. 3220–3243, 2014, Accessed: 00, 2020. [Online]. Available: