On The Number Of Topologies On A Finite Set

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime $p$, $T(p^k)\equiv k+1 \ (mod \ p)$, and that for each natural number $n$ there exists a unique $k$ such that $T(p+n)\equiv k \ (mod \ p)$. We calculate $k$ for $n=0,1,2,3,4$. We give an alternative proof for a result of Z. I. Borevich to the effect that $T_0(p+n)\equiv T_0(n+1) \ (mod \ p)$

Character free proofs for two solvability theorems due to Isaacs

We give character-free proofs of two solvability theorems due to Isaacs

A sufficient condition for fixed points of a coprime action to have a normal complement

Let A be a finite group acting on a finite group G via automorphisms. Assume that (|A|,|G|)=1. We prove that if CG(A) is a Hall -subgroup of G, then G has a normal -complement

Some sufficient conditions for p-nilpotency of a finite group

Let G be a finite group and let p be prime dividing . In this article, we supply some sufficient conditions for G to be p-nilpotent (see Theorem 1.2) as an extension of the main theorem of Li et al. (J. Group Theor. 20(1): 185-192, 2017)

Bazı alt grup yerleşme özelliklerinin bir sonlu grubun yapısı üzerine etkileri.

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 $Ph.D. - Doctoral Progra

Cyclic intersections and control of fusion

Let H be a subgroup of a finite group G, and suppose that H contains a Sylow p-subgroup P of G. Write N=NG(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = \mathbf{N}_{G}(H)$$\end{document}, and assume that the Sylow p-subgroups of H boolean AND Hg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \cap Hg$$\end{document} are cyclic for all elements g is an element of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \in G$$\end{document} not lying in N. We show that in this situation, N controls G-fusion in P