Hall Subgroups Research Articles

On a problem posed by S. Li and J. Liu

A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. The aim of this note is to prove that a group G has a Hall normally embedded subgroup of order |B| for each subgroup B of G if and only if G is soluble with nilpotent residual cyclic of square-free order. This is the answer to a problem posed by Li and Liu (J. Algebra 388:1–9, 2013).

ON SYLOW NORMALIZERS OF FINITE GROUPS

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

On Hall subnormally embedded and generalized nilpotent groups

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if H is a Hall subgroup of the subnormal closure HSG. In this paper, some new characterizations for finite groups are obtained based on the assumption that some subgroups are Hall subnormally embedded.

On Hall subgroups of a finite group

Abstract New criteria of existence and conjugacy of Hall subgroups of finite groups are given.

Induction and restriction of characters and Hall subgroups

Let H be a Hall π-subgroup of a π-separable group G. Given a character χ∈Irr(G), we determine when it happens that χ=αG for every irreducible constituent α of χH, and dually, given a character α∈Irr(H), we determine when it happens that α=χH for every irreducible constituent χ of αG. We also study some related questions.

Pronormality and Strong Pronormality of Subgroups

A subgroup H of a group G is said to be pronormal if, for any element g ∈ G, subgroups H and H g are conjugate in $$ \left\langle {H,{H^g}} \right\rangle $$ . A subgroup H of a group G is said to be strongly pronormal if, for any subgroup K ≤ H and any element g ∈ G, there exists an element $$ x\in \left\langle {H,{K^g}} \right\rangle $$ such that K gx ≤ H. Many known examples of pronormal subgroups, namely, normal subgroups, maximal subgroups, Sylow subgroups of finite groups, and Hall subgroups of finite soluble groups, will also exemplify strongly pronormal subgroups. It is shown that Carter subgroups of finite groups (which are always pronormal) are not strongly pronormal in general, even in soluble groups.

Sylow numbers and nilpotent Hall subgroups

Let π be a set of primes and G a finite group. We characterize the existence of a nilpotent Hall π-subgroup of G in terms of the number of Sylow subgroups for the primes in π.

Finite groups with nilpotent and Hall subgroups

- The structure of a finite group whose subgroups are all either nilpotent or Hall subgroups is established.

On the pronormality of hall subgroups

Fix a set of primes π. A finite group is said to satisfy C π or, in other words, to be a C π -group, if it possesses exactly one class of conjugate π-Hall subgroups. The pronormality of π-Hall subgroups in C π -groups is proven, or, equivalently, we show that C π is inherited by overgroups of π-Hall subgroups. Thus an affirmative solution is obtained to Problem 17.44(a) from The Kourovka Notebook. We also provide some example demonstrating that Hall subgroups in finite groups are not pronormal in general.

On X-Permutable Subgroups

A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that ABx=BxA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.

Groups with two Sylow numbers are the product of two nilpotent Hall subgroups

We have recently found a characterization of the existence of nilpotent Hall subgroups in a finite group. Using this result, we prove the statement in the title. This is a strong form of a conjecture of J. Zhang.

Finite groups with hall Schmidt subgroups

A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. We study the properties of a non-nilpotent group G in which every Schmidt subgroup is a Hall subgroup of G.

Permutation Groups Containing a Regular Abelian Hall Subgroup

We give a useful description of the permutation groups containing a regular abelian Hall subgroup.

Computing Hall subgroups of finite groups

AbstractWe describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

On Isomorphism Testing of Groups with Normal Hall Subgroups

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group H over \( \mathbb{Z}_p^d \), p a prime, determine if there exists an automorphism : H → H, such that the induced representation ρ𝜙 = ρ ◦ 𝜙 and τ are equivalent, in time poly(|H|, p d).

Pronormality of Hall subgroups in finite simple groups

We prove that Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of "Kourovka notebook".

SOME PROPERTIES OF HALL SUBGROUP

In the present paper , the order and index of the normal abelian Hall subgroups have been studied through using of some functions defined on group rings . Also , some properties of homomorphism have been studied .

Extending characters from Hall subgroups.

Suppose that G is a finite \pi -separable group. A classical result asserts that all irreducible characters of a Hall \pi -subgroup H of G extend to G if and only if H has a normal complement in G . Now, we fix a prime p and analyze when only the p' -degree irreducible characters of H extend to G .

New criterions of existence and conjugacy of Hall subgroups of finite groups

In the paper, new criterions for existence and conjugacy of Hall subgroups of finite groups are given. In particular, the Schur-Zassenhaus theorem, Hall theorem and Cunihin theorem are generalized.

On the number of classes of conjugate Hall subgroups in finite simple groups

In this paper we find the number of classes of conjugate π-Hall subgroups in all finite almost simple groups. We also complete the classification of π-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.