The influence of some embedding properties of subgroups onthe structure of a finite group

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2018

Öğrenci: MUHAMMET YASİR KIZMAZ

Danışman: GÜLİN ERCAN

Özet:

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 $