8 research outputs found
On The Number Of Topologies On A Finite Set
We denote the number of distinct topologies which can be defined on a set
with elements by . Similarly, denotes the number of distinct
topologies on the set . In the present paper, we prove that for any
prime , , and that for each natural number
there exists a unique such that . We calculate
for . We give an alternative proof for a result of Z. I.
Borevich to the effect that
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 , a subgroup is called a -subgroup if intersects trivially with distinct conjugates of itself. Suppose that is a Hall -subgroup of which is also a -subgroup. A famous theorem of Frobenius states that has a normal -complement whenever is self normalizing. In this case, is called a Frobenius complement and 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 be a -subgroup of which is also a Hall subgroup of . Then has a normal complement in if and only if has a normal complement in . Moreover, if is nonnormal in and has a normal complement in then is a Frobenius complement.} In the above configuration, the group need not be a Frobenius group, but the second part of the theorem guarantees the existence of a Frobenius group into which 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 -separable groups. This result shows that the structure of a -separable group admitting a Hall -subgroup which is also a -subgroup is very restricted. textbf{Theorem.}textit{ Let be a nonnormal -subgroup of the -separable group where is the set of primes dividing the order of . Further assume that is a Hall subgroup of . Then the following hold:} textit{ has -length where ;} textit{ there is an -invariant section of on which the action of is Frobenius. This section can be chosen as a chief factor of whenever is solvable;} textit{ is solvable if and only if is solvable and does not involve a subgroup isomorphic to .} 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 be a finite group having a cyclic Sylow -subgroup. Assume that every -subgroup of is abelian. Then is either -nilpotent or -closed.} textbf{Theorem.} textit{Let G be a finite group and let and 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}\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}\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}\end{document} not lying in N. We show that in this situation, N controls G-fusion in P