Henry Wilton
A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela.
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In this paper, we mainly investigate the converse of a well-known theorem proved by P. Hall, and present detailed characterizations under the various assumptions of the existence of some families of Hall subgroups. In particular, we prove that if $p\neq 3$ and a finite group $G$ has a Hall $\{p,q\}$-subgroup for every prime $q\neq p$, then $G$ is $p$-soluble.
On finite trifactorised groups and Sylow and Hall theorems for skew braces
本文证明Truman关于有限斜braces的Sylow和Hall定理是有限三因子化群Sylow和Hall结构的直接推论，并自然导出Cauchy定理。
Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
本文建立了有限斜辫环的Sylow第一定理的无条件类比，并推导出Cauchy定理的类比，同时证明了可解加法和乘法群的有限斜辫环的Hall定理存在性部分的类比。
On $σ$-solvable hypergroups and related Hall's Theorem
该研究构建 $σ$-可解超群理论，分析其性质，并将 Hall 定理推广到该类超群中。
We generalize Philip Hall's celebrated theorems on finite solvable groups to scheme theory. Our result is based on a series of results on hypergroups.
We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$.
Let $γ_i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper central series of a group $G$, respectively. P. Hall showed that if $γ_{i+1}(G)$ is finite then the index $|G:Z_{2i}(G)|$ is finite. We prove that the same result holds under the weaker hypothesis that $|γ_{i+1}(G):γ_{i+1}(G)\cap Z_i(G)|$ is finite.
We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.
The paper studies the minimum polynomial degrees of $p$-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the order of the element.
In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.
Let $G$ be a finite group and $P\in Syl_p(G)$. We denote the $k$'th term of the upper central series of $G$ by $Z_k(G)$ and the norm of $G$ by $Z^*(G)$. In this article, we prove that if for every tame intersection $P\cap Q$ such that $Z_{p-1}(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. For $p=2$, we sharpen our results by proving if for every tame intersection $P\cap Q$ such that $Z^*(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. We also obtain several corollaries which give suffici…
Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear grou…
Extensions of a theorem of P. Hall on indexes of maximal subgroups
本文推广 P. Hall 关于极大子群指数与有限群可解性的经典定理，证明在极大子群幂零或真非极大子群含于素数（平方）指数子群时，$G$ 仍可解。
A generalization of the Arad--Ward theorem on Hall subgroups
本文推广 Arad–Ward 定理，证明同时属于两类 Hall 子群群类的有限群也属于其素数集交对应的群类，并推出在所有适当大小素数子集上均含 Hall 子群时存在可解 Hall $π$-子群。
Let $ σ$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $σ$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's class…
A computable version of Hall's Harem Theorem and Geometric von Neumann Conjecture
本文提出Hall定理的可计算版本，并将其应用于非可去的可计算粗空间，从而获得几何von Neumann猜想的可计算版本。
A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5.
Hall complement numbers
受张纪平对霍尔数分类问题启发，钱国华提出霍尔补数分类问题。本文证明每个霍尔补数要么是\(1\)，要么是\(4k + 2\)（\(k\)为非负整数），解决了钱国华的问题。
A simultaneous variant of the Hall-Paige theorem and applications
该研究提出Hall-Paige定理的同时变体性质$\mathcal{P}_k$，证明有限群$G$对$k<p-1$满足该性质且界紧，还将其与拉丁方的公共横截等概念关联，拓展了群论与拉丁方的交叉研究。
A $σ$-McKay theorem for $π$-separable groups
本研究针对$π$-可分群证明了G. Navarro提出的McKay猜想的Hall $σ$-子群类比结论，该结论同时推广了经典McKay猜想及其$π$-可分版本，验证了相关不可约特征标计数等式。
In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox.
We prove a computable version of Hall's Harem Theorem and apply it to computable versions of Tarski's alternative theorem.
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