Nonsolvable Groups Research Articles

Two characterizations of the smallest non-solvable group

We introduce two invariants for finite groups, and use each to characterize A 5.

Harmonic mean Sylow numbers of nonsolvable groups

Harmonic mean Sylow numbers of nonsolvable groups

Finite Groups Whose All Proper Subgroups Have Only Irreducible Characters of Square-free Degrees

Lewis determined the structure of finite groups whose irreducible character odd-degrees are square-free. In this paper, we give an identification of the structure of non-solvable groups whose proper subgroups have only square-free degrees.

Element orders and codegrees of characters in non-solvable groups

Given a finite group G and an irreducible complex character χ of G, the codegree of χ is defined as the integer cod(χ)=|G:ker(χ)|/χ(1). It was conjectured by G. Qian in [16] that, for every element g of G, there exists an irreducible character χ of G such that cod(χ) is a multiple of the order of g; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle.

Construction of one-fixed-point actions on spheres of nonsolvable groups II

Construction of one-fixed-point actions on spheres of nonsolvable groups II

The degrees of non-monomial characters and solvability of a finite group

For a positive integer l and a finite group G, let , where denotes the set of non-monomial characters of G. In this paper, we prove that if , then G is solvable. In particular, we show that for a non-solvable group G, if , then G = MC is a central product of two subgroups M and C such that and G/M is an M-group. Among the other results, we arrive at the solvability of G by assuming .

Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

Let G be a finite group. Denoting by textrm{cd}(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the numbers in textrm{cd}(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in textrm{cd}(G). In the series of three papers starting with the present one, we analyze the structure of the finite non-solvable groups whose character degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph.

5-Regular prime graphs of finite nonsolvable groups

Abstract The prime graph Δ ⁢ ( G ) \Delta(G) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and 𝑞 are joined by an edge if the product p ⁢ q pq divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31] proposed the following conjecture. Let 𝐺 be a group and let k ≥ 5 k\geq 5 be odd. If the prime graph Δ ⁢ ( G ) \Delta(G) is 𝑘-regular, then Δ ⁢ ( G ) \Delta(G) is a complete graph of order k + 1 k+1 . In this paper, we show that if the prime graph Δ ⁢ ( G ) \Delta(G) of a finite nonsolvable group 𝐺 is 5-regular, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} or possibly the graph depicted in the first figure below. Moreover, if 𝐺 is an almost simple group, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} .

Finite groups with Hall normally embedded minimal subgroups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of H , where H is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called an HNE 2-group if all cyclic subgroups of order 2 and 4 of G are Hall normally embedded in G. In this paper, we prove that all HNE 2-groups are 2-nilpotent. Furthermore, we also characterize the structure of finite group all of whose maximal subgroups are HNE 2-groups. Finally, we determine finite non-solvable groups all of whose second maximal subgroups are HNE 2-groups.

Quadratic number fields with unramified SL2(5)-extensions

Continuing the line of thought of an earlier work [11], we provide the first infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group SL2(5), the (unique) smallest nonsolvable group for which this problem was previously open. Our approach also improves upon [11] by yielding the first infinite family of real-quadratic fields possessing an unramified Galois extension whose Galois group is perfect and not generated by involutions. Our result also amounts to a new existence result on quintic number fields with squarefree discriminant and additional local conditions.

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.

Non-solvable groups whose character degree graph has a cut-vertex. III

Let G be a finite group. Denoting by textrm{cd}(G) the set of the degrees of the irreducible complex characters of G, we consider the character degree graph of G: this, is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in textrm{cd}(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in textrm{cd}(G). This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 that these groups have a unique non-solvable composition factor S, and that S is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for S were treated, except the case Scong textrm{PSL}_{2}(2^a) for some integer age 2; the remaining case is addressed in the present paper.

On some second maximal subgroups of non-solvable groups

We call a group $G$ belongs to the class of groups $S_{p}'$, if for every $pd$-chief factor $A/B$ of $G$, $((A/B)_{p})'=1$. In this paper, we investigate the influence of some second maximal subgroups which are related to non-$c_{p}$-normal maximal subgroups on the structure of $S_{p}'$.

Classification of non-solvable groups whose power graph is a cograph

Abstract In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.

Non-solvable groups whose character degree graph has a cut-vertex. II

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph $$\Delta (G)$$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in $$\textrm{cd}(G)$$ , and two distinct vertices p, q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ . This paper continues the work, started in [8], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in [8], groups with no composition factors isomorphic to $$\textrm{PSL}_{2}(t^a)$$ (for any prime power $$t^a\ge 4$$ ) were treated, here we consider the complementary situation in the case when $$t$$ is odd and $$t^a> 5$$ . The proof of this classification will be then completed in the third and last paper of this series [7] that deals with the case $$t=2$$ .

Finite groups all of whose proper subgroups have few character values

<abstract><p>In this paper, the structures of non-solvable groups whose all proper subgroups have at most seven character values are identified.</p></abstract>

Groups with ALOGTIME-hard Word Problems and PSPACE-complete Compressed Word Problems

We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable groups. This class includes free groups, Grigorchuk’s group, and Thompson’s groups. We prove that these groups have an NC 1 -hard word problem and that for some of them (including Grigorchuk’s group and Thompson’s groups) the compressed word problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.

Measurement as a Shortcut to Long-Range Entangled Quantum Matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

Nonsolvable groups have a large proportion of vanishing elements

Nonsolvable groups have a large proportion of vanishing elements

Groups with some cube-free irreducible character co-degrees

For a character χ of a finite group G, the number is called the co-degree of χ. Let be an integer and denote the set of irreducible characters whose kernels do not contain . In this paper, we show that if G is solvable and for every prime divisor p of and every , then the derived length of G is at most . Then, we classify the finite non-solvable groups with non-trivial Fitting subgroups such that the co-degrees of their irreducible characters whose kernels do not contain the Fitting subgroups are cube-free.