Centralizers of Finite p-Subgroups in Simple Locally Finite Groups

We are interested in the following questions of B. Hartley: (1) Is it true that, in an infinite, simple locally finite group, if the centralizer of a finite subgroup is linear, then G is linear? (2) For a finite subgroup F of a non-linear simple locally finite group is the order vertical bar CG(F)vertical bar infinite? We prove the following: Let G be a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups. Let p be a fixed prime and s is an element of N. Then for any finite p subgroup F of G, the centralizer C-G(F) contains subgroups isomorphic to the homomorphic images of SL(s, F-q). In particular C-G(F) is a non-linear group. We also show that if F is a finite p-subgroup of the infinite locally finite simple group G of classical type and given s is an element of N and the rank of G is sufficiently large with respect to vertical bar F vertical bar and s, then C-G(F) contains subgroups which are isomorphic to homomorphic images of SL(s, K).


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Let A be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal I subset of A, Drinfeld defined the notion of structure of level I on a Drinfeld module.
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Öztürk, Semra (Cambridge University Press (CUP), 2003-04-01)
Let G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set X-Cn and the multiplicities of indecomposable summands of M considered as a kC(n)-module are related via a localization theorem in equivariant cohomology, where C-n is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.
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Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p (n) q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.
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GÜRSES, METİN; Pekcan, Asli; Zheltukhın, Kostyantyn (Elsevier BV, 2020-01-31)
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
Chirality of real non-singular cubic fourfolds and their pure deformation classification
Finashin, Sergey (Springer Science and Business Media LLC, 2020-02-22)
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.
Citation Formats
M. Kuzucuoğlu, “Centralizers of Finite p-Subgroups in Simple Locally Finite Groups,” JOURNAL OF SIBERIAN FEDERAL UNIVERSITY-MATHEMATICS & PHYSICS, pp. 281–286, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/43775.