Supersolvable Groups Research Articles

Finite supersolvable groups and Hall normally embedded subgroups of prime power order

Finite supersolvable groups and Hall normally embedded subgroups of prime power order

The pro-supersolvable topology on a free group: Deciding denseness

Let F be a free group of arbitrary rank and let H be a finitely generated subgroup of F. Given a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Our main result states that it is decidable whether H is pro-Su dense, where Su⊂S denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether H is pro-S dense?

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

Groups with fewer than 15 involutions

We show that if G is a finite group with fewer than 15 involutions, then . We then show that if G contains fewer than 15 involutions such that , then G is a solvable group of 2-rank 1. Furthermore, we show that if G contains involutions and , then G is a supersolvable group, but it is not necessarily true for . Finally, we show that a non-solvable group generated by involutions has at least 15 involutions.

Nilary group rings and algebras

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions for the group ring $A[G]$ to be (principally) nilary in terms of conditions on the ring $A$ or the group $G$. For example, we show that: (1) If $A[G]$ is (pr-)nilary, then $A$ is (pr-)nilary and either $G$ is prime or the order of each finite nontrivial normal subgroup of $G$ is nilpotent in $A$. (2) Assume that $G$ is finite. Then $G$ is nilpotent and $A[G]$ is (pr-)nilary if and only if $G$ is a $p$-group, $char(A)=p^\alpha $ ($p$ is a prime), and $A$ is (pr-)nilary. (3) Let $G$ be a finite supersolvable group such that $q$ is the smallest prime dividing $ G ,$ and $F$ is a field with $char(F)=q$. Then $F[G]$ is nilary if and only if $G$ is a $q$-group. Examples are provided to illustrate and delimit our results.

On finite groups with given ICΦ-subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text]. We analyze the structure of a finite group [Formula: see text] under the assumption that some given subgroups of [Formula: see text] are [Formula: see text]-subgroups of [Formula: see text]. A new characterization of finite abelian groups and some new criteria for [Formula: see text]-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.

On the Finite Group Which Is a Product of Two Subnormal Supersolvable Subgroups

Let G be a finite group that is a product of two subnormal ( normal) supersolvable subgroups. The following are interesting topics in the study of the structure of G: obtaining the conditions in addition to guarantee that G is supersolvable and giving the detailed structure of G when G is non-supersolvable. In this paper, we obtain a characteristic property of G being non-supersolvable and two new sufficient conditions for G being a supersolvable group.

Deterministic algorithms for the hidden subgroup problem

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a~$\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected\/} query complexity that is sensitive to the input, namely~$ \Order(\sqrt{ n/m}\, )$, where~$m$ is the order of the hidden subgroup. In the first version of this article~\cite[Sec.~5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li~\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian\/} groups which solve the problem with~$ \Order(\sqrt{ n/m }\, )$ queries, and find the hidden subgroup with~$ \Order( \sqrt{ n (\log m) / m} + \log m ) $ queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~$\order(\sqrt{ n/m } \,)$, and that of \emph{finding\/} the entire subgroup may also be~$\order(\sqrt{ n/m } \,)$ or even~$\upomega(\sqrt{ n/m } \,) $.}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~$ \Order(\sqrt{ n/m }\, )$ for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~$ \Order(\sqrt{ (n/m) \log (n/m) }\,) $ for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~$\log m$ multiplicative factor increase in the query complexity.

Bm$p$-supersolvable fusion systems

In this paper, we consider finite groups. Let $p$ be a prime and $S$ be a $p$-group. Let $\mathcal~{F}$ be a saturatedfusion system over $S$. We first define $p$-supersolvable fusion systems. Then we prove that the models of $p$-supersolvablefusion systems are $p$-supersolvable groups and give the $p$-supersolvablity of normal subsystems and factors of a $p$-supersolvable fusion system. Last, wegive the criterion of $p$-supersolvable fusion systems.

Finite groups with three nonabelian subgroups

We characterize finite groups with exactly two nonabelian proper subgroups. When $G$ is nilpotent, we show that $G$ is either the direct product of a minimal nonabelian $p$-group and a cyclic $q$-group or a $2$-group. When $G$ is nonnilpotent supersolvable group, we obtain the presentation of $G$. Finally, when $G$ is nonsupersolvable, we show that $G$ is a semidirect product of a $p$-group and a cyclic group.

On NH-embedded subgroups of finite groups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. [Formula: see text] is said to be [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a Hall subgroup of [Formula: see text] and [Formula: see text], where [Formula: see text] is the largest [Formula: see text]-semipermutable subgroup of [Formula: see text] contained in [Formula: see text]. In this paper, we give some new characterizations of [Formula: see text]-nilpotent and supersolvable groups by using [Formula: see text]-embedded subgroups. Some known results are generalized.

On Π-property of subgroups of a finite group

Let [Formula: see text] be a finite group, [Formula: see text] a prime, [Formula: see text] a Sylow [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] a power of [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the unique smallest normal subgroup of [Formula: see text] for which the corresponding factor group is abelian of exponent dividing [Formula: see text]. Let [Formula: see text], [Formula: see text], [Formula: see text] be classes of all [Formula: see text]-groups, [Formula: see text]-nilpotent groups and [Formula: see text]-supersolvable groups, respectively, [Formula: see text] be the [Formula: see text]-residual of [Formula: see text]. Let [Formula: see text]. A subgroup [Formula: see text] of a finite group [Formula: see text] is said to have [Formula: see text]-property in [Formula: see text], if for any [Formula: see text]-chief factor [Formula: see text], [Formula: see text] is a [Formula: see text]-number. A normal subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-hypercyclically embedded in [Formula: see text] if every [Formula: see text]-[Formula: see text]-chief factor of [Formula: see text] is cyclic, where [Formula: see text] is a fixed prime. In this paper, we prove that [Formula: see text] is [Formula: see text]-hypercyclically embedded in [Formula: see text] if and only if for some [Formula: see text]-subgroups [Formula: see text] of [Formula: see text], [Formula: see text] have [Formula: see text]-property in [Formula: see text].

Basic results on an unsolved problem about factorization of finite groups

We deal with an unsolved problem in group theory (question 19.35 of Kourovka Notebook). It asks whether, given a finite group G and a factorization one can always find subsets such that and The case k = 2 is of special interest. We prove that the answer is positive for all finite supersolvable (and so abelian) groups. However, G. Bergman has recently given a counterexample to the question for the case k = 3. Regarding the special case k = 2, at first, we prove that the statement is true for all finite solvable groups, and groups such that for every divisor d of their order, there is a subgroup of order or index d. Thereafter, by proving an extension theorem we show that the answer is positive if it is true for all simple groups with no subgroup of prime index. Also, we give some other theorems and criteria, for checking the property and state a question about some candidates for probable counterexamples. We end by noting some related connections, questions and problems about the topic.

On two conjectures about the sum of element orders

AbstractLet G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$ , where $o(g)$ denotes the order of $g \in G$ . There are many results on the influence of this function on the structure of a finite group G.In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and $\psi (G)>31\psi (C_n)/77$ , where $C_n$ is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$ , then $G\cong A_4 \times C_m$ , where $(m, 6)=1$ .Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$ , then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$ . By an example, we show that this conjecture is not satisfied in general.

π-Quasinormally embedded subgroups and p-nilpotency of finite groups

Let P be a nontrivial p-group. A chain of subgroups is called a maximal chain of P provided that In this article, we determine the structure of finite groups having π-quasinormally embedded subgroups of a maximal chain of Sylow subgroups. We obtain new characterizations of finite p-nilpotent and supersolvable groups.

Finite Groups with $$\mathfrak F$$-Subnormal Subgroups

Let $$G$$ be a finite group, let $$M$$ be a maximal subgroup of $$G$$ , and let $$\mathfrak F$$ be a hereditary formation consisting of solvable groups. The metanilpotency of the $$\mathfrak F$$ -residual $$G^\mathfrak F$$ is established under the assumption that all subgroups maximal in $$M$$ are $$\mathfrak F$$ -subnormal in $$G$$ , and the nilpotency of $$G^\mathfrak F$$ is established in the case where $$\mathfrak F$$ is saturated. Properties of the group $$G$$ are indicated in more detail for the formation of all solvable groups with Abelian Sylow subgroups, for the formation of all supersolvable groups, and for the formation of all groups with nilpotent commutator subgroup.

On the Zassenhaus Conjecture for Certain Cyclic-by-Nilpotent Groups

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group G is conjugate within the rational group algebra to an element of the form $$\pm g$$ with $$g\in G$$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However, it is known to be true for many classes of solvable groups, as for example nilpotent groups, cyclic-by-abelian groups, and groups having a Sylow subgroup with abelian complement. On the other hand, the conjecture remains open for the class of supersolvable groups. This paper is a contribution to this question. More precisely, we study the conjecture for the class of cyclic-by-nilpotent groups with a special attention to the class of cyclic-by-Hamiltonian groups. We prove the conjecture for cyclic-by-p-groups and some cyclic-by-Hamiltonian groups.

The permutability of p-sylowizers of some p-subgroups in finite groups

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. The main aim of this paper is to investigate the influence of p-sylowizers on the structure of finite groups. We obtained some new characterizations of p-nilpotent and supersolvable groups by the permutability of the p-sylowizers of some p-subgroups. In addition, we determined all p-sylowizers of arbitrary p-subgroups for the supersolvable groups.

A New Algorithm for Fast Generalized DFTs

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups G . The new algorithm uses O (∣ G ∣ ω /2 + o (1) ) operations to compute the generalized DFT over finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 2. Previously, “exponent one” algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known, even under the assumption ω = 2, for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for SL 2 (F q ) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group and exponent one if ω = 2. Our algorithm also yields an improved exponent for computing the generalized DFT over general finite groups G , which beats the longstanding previous best upper bound for any ω. In particular, assuming ω = 2, we achieve exponent √ 2, while the previous best was 3/2.

On supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal

A finite group G is called an MSN∗-group if it is supersolvable, and all maximal subgroups of the Sylow subgroups of G are subnormal in G. A group G is called a minimal non-MSN∗-group if every proper subgroup of G is an MSN∗-group but G itself is not. In this paper, we obtain a complete classification of minimal non-MSN∗-groups.