In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by Betz and Nash. We also provide an application of our results in studying comaximal subgroup graph of a group. Finally, we conclude with some open issues.
For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$.
Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $σ_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $σ_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$. This solves an open problem posed in \cite{8}.
The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.
We exhibit infinitely many natural numbers $n$ for which there exists at least one insolvable group of order $n$, and yet the holomorph of any solvable group of order $n$ has no insolvable regular subgroup. We also solve Problem 19.90 (d) in the Kourovka notebook.
In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.
We prove a new criterion for the solvability of the finite groups, depending on the function $ψ_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability of some groups for which the solvability does not follow from earlier similar kind of results and we emphasize the following: looking at $ψ_k(G)$ for $k>1$ can be useful to get further pieces of information about the group $G$.
A Criterion for Solvability of a Finite Group by the Sum of Element Orders
设$G$是一个有限群，有$ψ(G) = \sum_{g \in G} o(g)$，其中$o(g)$表示$g \in G$的阶.在[M. Herzog，et.例如，有限群可解性的两个新判据，J. Algebra，2018]，作者提出了如下猜想：\textbf{Conjecture.} \textit{If $G$ is a group of order $n$ and $ψ(G)>211ψ(C_n)/1617 $, where $C_n$ is the cyclic group of order $n$, then $G$ is solvable.}本文证明了这个猜想的有效性。
Let G be a finite group. Denote by ψ(G) the sum ψ(G)=\sum_{x\in G}|x| where |x| denotes the order of the element x, and by o(G) the quotient o(G)=\frac{ψ(G)}{|G|}. Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M. Zarrin, we prove that if o(G)< o(A_5), then G is solvable.
For a knot $K$ in $S^3$, let $G(K)$ be the knot group of $K$, $a(K)$ the Ma-Qiu index (the MQ index, for short), which is the minimal number of normal generators of the commutator subgroup of $G(K)$, and $m(K)$ the Nakanishi index of $K$, which is the minimal number of generators of the Alexander module of $K$.We generalize the notions for a pair of a group $G$ and its normal sugroup $N$, and we denote them by $a(G, N)$ and $m(G, N)$ respectively.Then it is easy to see $m(G, N)\le a(G, N)$ in general.We also introduce a notion ``$ω$-solvability" for a group that the intersection of all higher …
We give a new criterion for solvability of group equations, providing proofs of various generalizations of the Kervaire-Laudenbach conjecture for Connes-embeddable groups.
Group Structure from Subgroup and Cyclic Subgroup Counts
Presidency University（总统大学）；Indian Institute of Technology, Jammu（印度技术学院贾姆分校）；Universidad de los Andes（洛斯安第斯大学）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究学院贝尔哈姆普尔分校）
In this note, we provide a (super-exponential time) algorithm to solve the generalized conjugacy problem in relatively hyperbolic groups, given solvability of the generalized conjugacy problem in each of the parabolic subgroups.
We investigate partial Equality and Word Problems for finitely generated groups. After introducing Upper Banach (UB) density on free groups, we prove that solvability of the Equality Problem on squares of UB-generic sets implies solvability of the whole Word Problem. In particular, we prove that solvability of generic EP implies WP. We then exploit another definition of generic EP, which turns out to be equivalent to generic WP. We characterize in different ways the class of groups with unsolvable UB-generic WP, proving that it contains that of algorithmically finite groups, and it is containe…
We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.
Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{ε_{1}}g_{2}t^{ε_{2}} \cdots g_{n}t^{ε_{n}}=1 \; (g_{i} \in G,\ ε_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} \in G$. In this paper we show that $s(t)=1$ has a solution over $G$ provided a single relation on coefficients of $s(t)$ holds. We also generalize our results to equations containing higher powers of $t$. The later equations are also related to Kaplansky zero-divisor conjecture \cite{K}.
In the recent paper [A generalization of Taketa's theorem on M-groups, Quaestiones Mathematicae, (2022), https://doi.org/10.2989/16073606.2022.2081632], we give an upper bound 5/2 for the average of non-monomial character degrees of a finite group G, denoted by acdnm(G), which guarantees the solvability of G. Although the result is true, the example we gave to show that the bound is sharp turns out to be incorrect. In this paper, we find a new bound and we give an example to show that this new bound is sharp. Indeed, we prove the solvability of G, by assuming acdnm(G) < acdnm(SL2(5)) = 19/7.
We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide …
