The Subgroup Structure of Finite Alternating and Symmetric Groups
نویسندگان
چکیده
In this course we will be studying the subgroup structure of the finite alternating and symmetric groups. What does the phrase “study the subgroups of symmetric groups” mean? In this introduction I’ll suggest an answer to that question, and attempt to convince you that answer has some merit. In the process you’ll get some idea of the material we will be covering, and I’ll attempt to motivate that material. First, given a finite group G and a set Ω, define a permutation representation of G on Ω to be a homomorphism π : G → S = Sym(Ω) of G into the symmetric group on Ω. From one point of view, the study of the subgroup structure of S amounts to the study of such representations, since the subgroups of S are precisely the images of these representations. At first glance it is not clear that this restatement of the problem represents any progress, as it would appear to be equally vague. However in awhile we will see that the reformulation does have some advantages. The theory of permutation representations can be embedded in a much more general representation theory of groups, and the first few sections of the notes discuss that theory. In particular in any representation theory, we will wish to take advantage of two types of reductions: First, reduce the study of the general representation of G to the study of the indecomposable and irreducible representations of G. Second, reduce the study of representations of the general finite group to the study of representations of almost simple groups, where G is almost simple if its generalized Fitting subgroup is a nonabelian simple group. (ie. G has a unique minimal normal subgroup, and that subgroup is a nonabelian simple group.)
منابع مشابه
On groups and initial segments in nonstandard models of Peano Arithmetic
This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducin...
متن کاملThe influence of S-embedded subgroups on the structure of finite groups
Let H be a subgroup of a group G. H is said to be S-embedded in G if G has a normal T such that HT is an S-permutable subgroup of G and H ∩ T ≤ H sG, where H denotes the subgroup generated by all those subgroups of H which are S-permutable in G. In this paper, we investigate the influence of minimal S-embedded subgroups on the structure of finite groups. We determine the structure the finite grou...
متن کاملOn $Phi$-$tau$-quasinormal subgroups of finite groups
Let $tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$. Let $bar{G}=G/H_{G}$ and $bar{H}=H/H_{G}$. We say that $H$ is $Phi$-$tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $bar{T}$ of $bar{G}$ and some $tau$-subgroup $bar{S}$ of $bar{G}$ contained in $bar{H}$, $bar{H}bar{T}$ is $S$-quasinormal in $bar{G}$ and $bar{H}capbar{T}leq bar{S}Phi(bar{H})$. I...
متن کاملON p-NILPOTENCY OF FINITE GROUPS WITH SS-NORMAL SUBGROUPS
Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...
متن کامل