Show/Hide Menu
Hide/Show Apps
Logout
Türkçe
Türkçe
Search
Search
Login
Login
OpenMETU
OpenMETU
About
About
Open Science Policy
Open Science Policy
Communities & Collections
Communities & Collections
Help
Help
Frequently Asked Questions
Frequently Asked Questions
Guides
Guides
Thesis submission
Thesis submission
MS without thesis term project submission
MS without thesis term project submission
Publication submission with DOI
Publication submission with DOI
Publication submission
Publication submission
Supporting Information
Supporting Information
General Information
General Information
Copyright, Embargo and License
Copyright, Embargo and License
Contact us
Contact us
The Influence of some embedding properties of subgroups on the structure of a finite group
Download
index.pdf
Date
2018
Author
Kızmaz, Muhammet Yasir
Metadata
Show full item record
Item Usage Stats
347
views
60
downloads
Cite This
In a finite group $G$, a subgroup $H$ is called a $TI$-subgroup if $H$ intersects trivially with distinct conjugates of itself. Suppose that $H$ is a Hall $pi$-subgroup of $G$ which is also a $TI$-subgroup. A famous theorem of Frobenius states that $G$ has a normal $pi$-complement whenever $H$ is self normalizing. In this case, $H$ is called a Frobenius complement and $G$ is said to be a Frobenius group. A first main result in this thesis is the following generalization of Frobenius' Theorem. textbf{Theorem.}textit{ Let $H$ be a $TI$-subgroup of $G$ which is also a Hall subgroup of $N_G(H)$. Then $H$ has a normal complement in $N_G(H)$ if and only if $H$ has a normal complement in $G$. Moreover, if $H$ is nonnormal in $G$ and $H$ has a normal complement in $N_G(H)$ then $H$ is a Frobenius complement.} In the above configuration, the group $G$ need not be a Frobenius group, but the second part of the theorem guarantees the existence of a Frobenius group into which $H$ can be embedded as a Frobenius complement. Another contribution of this thesis is the following theorem, which extends a result of Gow (see Theorem ref{int gow}) to $pi$-separable groups. This result shows that the structure of a $pi$-separable group admitting a Hall $pi$-subgroup which is also a $TI$-subgroup is very restricted. textbf{Theorem.}textit{ Let $H$ be a nonnormal $TI$-subgroup of the $pi$-separable group $G$ where $pi$ is the set of primes dividing the order of $H$. Further assume that $H$ is a Hall subgroup of $N_G(H)$. Then the following hold:} textit{$a)$ $G$ has $pi$-length $1$ where $G=O_{pi'}(G)N_G(H)$;} textit{$b)$ there is an $H$-invariant section of $G$ on which the action of $H$ is Frobenius. This section can be chosen as a chief factor of $G$ whenever $O_{pi'}(G)$ is solvable;} textit{$c)$ $G$ is solvable if and only if $O_{pi'}(G)$ is solvable and $H$ does not involve a subgroup isomorphic to $SL(2,5)$.} In the last chapter we focus on giving alternative proofs without character theory for the following two solvability theorems due to Isaacs (cite{isa3}, Theorem 1 and Theorem 2). Our proofs depend on transfer theory and graph theory. textbf{Theorem.} textit{Let $G$ be a finite group having a cyclic Sylow $p$-subgroup. Assume that every $p'$-subgroup of $G$ is abelian. Then $G$ is either $p$-nilpotent or $p$-closed.} textbf{Theorem.} textit{Let G be a finite group and let $pneq 2$ and $q$ be primes dividing $
Subject Keywords
Finite groups.
,
Group theory.
,
TI subgroups.
,
Frobenius groups.
URI
http://etd.lib.metu.edu.tr/upload/12622530/index.pdf
https://hdl.handle.net/11511/27764
Collections
Graduate School of Natural and Applied Sciences, Thesis
Suggestions
OpenMETU
Core
Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms
Ercan, Gülin; Khukhro, E. I. (2014-07-01)
A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with certain additional restrictions (which are satisfied, in particular, if either |FH| is odd or |H| = 2). In the case where G is a finite p-group such that G = [G, F] it is proved that the rank of G is bounded ab...
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...
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
On the nilpotent length of a finite group with a frobenius group of automorphisms
Öğüt, Elif; Ercan, Gülin; Güloğlu, İsmail Ş.; Department of Mathematics (2013)
Let G be a finite group admitting a Frobenius group FH of automorphisms with kernel F and complement H. Assume that the order of G and FH are relatively prime and H acts regularly on the fixed point subgroup of F in G. It is proved in this thesis that the nilpotent length of G is less than or equal to the sum of the nilpotent length of the commutator group of G and F with 1 and the nilpotent length of the commutator group of G and F is equal to the nilpotent length of the fixed point subgroup of H in the co...
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.
Citation Formats
IEEE
ACM
APA
CHICAGO
MLA
BibTeX
M. Y. Kızmaz, “The Influence of some embedding properties of subgroups on the structure of a finite group,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.