kappa-Existentially closed groups

Date

2018-04-01

Author

Kegel, Otto H.
Kuzucuoğlu, Mahmut

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

180
views

0
downloads

Let kappa is be an infinite cardinal. The class of kappa-existentially closed groups is defined and their basic properties are studied. Moreover, for an uncountable cardinal kappa, uniqueness of kappa-existentially closed groups are shown, provided that they exist. We also show that for each regular strong limit cardinal kappa, there exists kappa-existentially closed groups. The structure of centralizers of subgroups of order less than kappa in a kappa-existentially group G are determined up to isomorphism namely, for any subgroup F <= G(nu) in G with vertical bar F vertical bar < kappa, the subgroup C-G(F) is isomorphic to an extension of Z(F) by G.

Subject Keywords

Infinite symmetric groups, Centralizers of subgroups, Existentially closed groups, Universal groups

URI

https://hdl.handle.net/11511/47026

Journal

JOURNAL OF ALGEBRA

DOI

https://doi.org/10.1016/j.jalgebra.2017.12.005

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

On the existence of kappa-existentially closed groups Kegel, Otto H.; Kaya, Burak; Kuzucuoğlu, Mahmut (2018-09-01) We prove that a κ-existentially closed group of cardinality λ exists whenever κ ≤ λ are uncountable cardinals with λ^{<κ} = λ. In particular, we show that there exists a κ-existentially closed group of cardinalityκ for regular κ with 2^{<κ} = κ. Moreover, we prove that there exists noκ-existentially closed group of cardinality κ for singular κ. Assuming thegeneralized continuum hypothesis, we completely determine the cardinalsκ ≤ λ for which a κ-existentially closed group of cardinality λ exists
An infinite family of strongly real Beauville p-groups Gul, Sukran (2018-04-01) We give an infinite family of non-abelian strongly real Beauville p-groups for every prime p by considering the quotients of triangle groups, and indeed we prove that there are non-abelian strongly real Beauville p-groups of order for every or 7 according as or or . This shows that there are strongly real Beauville p-groups exactly for the same orders for which there exist Beauville p-groups.
Centralizers of finite subgroups in Hall's universal group Kegel, Otto H.; Kuzucuoğlu, Mahmut (2017-01-01) The structure of the centralizers of elements and finite abelian subgroups in Hall's universal group is studied by B. Hartley by using the property of existential closed structure of Hall's universal group in the class of locally finite groups. The structure of the centralizers of arbitrary finite subgroups were an open question for a long time. Here by using basic group theory and the construction of P. Hall we give a complete description of the structure of centralizers of arbitrary finite subgroups in Ha...
Prime graphs of solvable groups Ulvi , Muhammed İkbal; Ercan, Gülin; Department of Electrical and Electronics Engineering (2020-8) If $G$ is a finite group, its prime graph $Gamma_G$ is constructed as follows: the vertices are the primes dividing the order of $G$, two vertices $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$. This thesis is mainly a survey that gives some important results on the prime graphs of solvable groups by presenting their proofs in full detail.
Fixed point free action on groups of odd order Ercan, Gülin; Güloğlu, İsmail Ş. (Elsevier BV, 2008-7) Let A be a finite abelian group that acts fixed point freely on a finite (solvable) group G. Assume that |G| is odd and A is of squarefree exponent coprime to 6. We show that the Fitting length of G is bounded by the length of the longest chain of subgroups of A.

Citation Formats

O. H. Kegel and M. Kuzucuoğlu, “kappa-Existentially closed groups,” JOURNAL OF ALGEBRA, pp. 298–310, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/47026.