On the existence of kappa-existentially closed groups

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


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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.
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 involutions in locally finite-simple groups
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We consider infinite locally finite-simple groups (that is, infinite groups in which every finite subset lies in a finite simple subgroup). We first prove that in such groups, centralizers of involutions either are soluble or involve an infinite simple group, and we conclude that in either case centralizers of involutions are not inert subgroups. We also show that in such groups, the centralizer of an involution is linear if and only if the ambient group is linear.
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Erdem, Fuat; Ercan, Gülin; Maróti, Attila; Department of Mathematics (2018)
Dixon showed that the probability that a random pair of elements in the symmetric group $S_n$ generates $S_n$ or the alternating group $A_n$ tends to $1$ as $n to infty$. (A generalization of this result was given by Babai and Hayes.) The generating graph $Gamma(G)$ of a finite group $G$ is defined to be the simple graph on the set of non-identity elements of $G$ with the property that two elements are connected by and edge if and only if they generate $G$. The purpose of this thesis is to study the graphs ...
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.
Citation Formats
O. H. Kegel, B. Kaya, and M. Kuzucuoğlu, “On the existence of kappa-existentially closed groups,” ARCHIV DER MATHEMATIK, pp. 225–229, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/37032.