Maximal Subgroup Research Articles

On the Fischer matrices of a group of shape 21+2n + :G

In this paper, the Fischer matrices of the maximal subgroup G = 21+8+ : (U4(2):2) of U6(2):2 will be derived from the Fischer matrices of the quotient group Q = G/Z(21+8+) = 28 : (U4(2):2), where Z(21+8+) denotes the center of the extra-special 2-group 21+8+. Using this approach, the Fischer matrices and associated ordinary character table of G are computed in an elegantly simple manner. This approach can be used to compute the ordinary character table of any split extension group of the form 21+2n+ :G, n ∈ N, provided the ordinary irreducible characters of 21+2n+ extend to ordinary irreducible characters of its inertia subgroups in 21+2n+:G and also that the Fischer matrices M(gi) of the quotient group 21+2n+ :G/Z(21+2n+) = 22n:G are known for each class representative gi in G.

Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform

For a Hermitian Lie group G, we study the family of representations induced from a character of the maximal parabolic subgroup P=MAN whose unipotent radical N is a Heisenberg group. Realizing these representations in the non-compact picture on a space I(ν) of functions on the opposite unipotent radical N¯, we apply the Heisenberg group Fourier transform mapping functions on N¯ to operators on Fock spaces. The main result is an explicit expression for the Knapp–Stein intertwining operators I(ν)→I(−ν) on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein–Sato identity for the Knapp–Stein kernel on N‾ and the decomposition of the metaplectic representation under the non-compact group M.

A note on finite groups in which every non-nilpotent maximal subgroup has prime index

Let [Formula: see text] be a finite group in which every non-nilpotent maximal subgroup has prime index and [Formula: see text] the largest prime divisor of [Formula: see text], without using the solvability of [Formula: see text] we prove that either the Sylow [Formula: see text]-subgroup of [Formula: see text] is normal or [Formula: see text] has a normal [Formula: see text]-complement.

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 $S_{p}'$.

Consistency of chiral symmetry breaking in chiral Yang-Mills theory with adiabatic continuity

We study the pattern of chiral symmetry breaking in the $\ensuremath{\psi}\ensuremath{\chi}\ensuremath{\eta}$ model [with the chiral fermion sector containing ${\ensuremath{\psi}}^{{ij}}$, ${\ensuremath{\chi}}_{[ij]}$, and ${\ensuremath{\eta}}_{i}^{A}$; see Armoni and Shifman Phys. Rev. D 85, 105003 (2012)] on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}_{L}^{1}$ and derive implications forcing ${\mathbb{R}}^{4}$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$, i.e., at weak coupling, no phase transitions are expected in passing to large $L({S}^{1})\ensuremath{\gg}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ (here, $\mathrm{\ensuremath{\Lambda}}$ is the dynamical Yang-Mills scale). Starting with the small-$L$ limit, we find the leading-order nonperturbative corrections in the given theory. The instanton-monopole operators induce the adjoint chiral condensate $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ at weak coupling, i.e., at $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$. Then adiabatic continuity tells us that $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ exists on ${\mathbb{R}}^{4}$, in full accord with the prediction from Bolognesi et al. [Phys. Rev. D 97, 094007 (2018)]. Simultaneously with $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{3}{\ensuremath{\delta}}_{k}^{i}$, the $\mathrm{SU}({N}_{c})$ gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.

MSGCorep: A package for corepresentations of magnetic space groups

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic (type-I, -III, and -IV MSGs) and nonmagnetic (type-II MSGs) materials. Program summaryProgram title:MSGCorepCPC Library link to program files:https://doi.org/10.17632/zt9r8pp8kr.1Developer's repository link:https://github.com/goodluck1982/MSGCorepLicensing provisions: GNU General Public Licence 3.0Programming language: WolframExternal routines/libraries used:SpaceGroupIrep (https://github.com/goodluck1982/SpaceGroupIrep)Nature of problem: The package MSGCorep provides offline database and tools for easy access to 1651 magnetic space groups, 122 magnetic point groups, and their corepresentations (coreps). MSGCorep is the first package that is able to calculate the direct product of full coreps for any magnetic space group and able to determine small coreps of energy bands for general purpose.Solution method: Coreps of a magnetic group are constructed from the representations of the maximal unitary subgroup of the magnetic group. Based on the complete representation data of 230 space groups provided by SpaceGroupIrep package, complete corep data of 1651 magnetic space groups are derived in MSGCorep package. A more convenient equation of induced corep, i.e. Eq. (2), is introduced to calculate the full coreps of all MSGs.

Finite groups with absolutely $\mathfrak{F}$-subnormal maximal subgroups

MSC: 20D10; 20E28 DOI: 10.21538/0134-4889-2023-29-1-254-258 This work was supported by the Ministry of Education of the Republic of Belarus (Grant number 20211467). A subgroup $M$ of a group $G$ is an $n$-maximal subgroups of $G$ if there is a subgroup chain $M=M_n\leq M_{n-1}\leq \ldots \leq M_1\leq M_0=G$ such that $M_{i+1}$ is a maximal subgroup of $M_i$. We establish a

P-characters of PSL(2,q)

For a finite group G, an irreducible character of G is called a P-character, if it is a constituent of permutation character (1M)G for some maximal subgroup M of G. In this article, we determine the P-characters of group PSL(2,q), and we check which of the permutation characters (1M)PSL(2,q) are multiplicity-free.

Conjugacy classes of maximal cyclic subgroups

Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

Character degrees of normally monomial 𝑝-groups of maximal class

AbstractA finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

Momentum maps and the Kähler property for base spaces of reductive principal bundles

AbstractWe investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup on the total space, and the Kähler property of special equivariant compactifications. We provide many examples illustrating that the main result is optimal.

On some products of finite groups

AbstractA classical result of Baer states that a finite group G which is the product of two normal supersoluble subgroups is supersoluble if and only if Gʹ is nilpotent. In this article, we show that if G = AB is the product of supersoluble (respectively, w-supersoluble) subgroups A and B, A is normal in G and B permutes with every maximal subgroup of each Sylow subgroup of A, then G is supersoluble (respectively, w-supersoluble), provided that Gʹ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms \mathcal{A}_m of an m -ary tree \mathcal{T}_m by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s\geq 1 orbits. We prove that such a group A embeds in a self-similar abelian group A^* which is also a maximal abelian subgroup of \mathcal{A}_m . The construction of A^{*} is based on the definition of a free monoid \Delta of rank s of partial diagonal monomorphisms of \mathcal{A}_m . Precisely, A^{*} = \overline{\Delta(B(A))} , where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of \mathcal{T}_m , and bar denotes the topological closure. Furthermore, we prove that if A is non-trivial, then A^{*} = C_{\mathcal{A}_m} (\Delta(A)) , the centralizer of \Delta(A) in \mathcal{A}_m . When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also \Delta -invariant for s=2 . In the final section, we introduce for m=ns \geq 2 , a generalized adding machine a , an automorphism of \mathcal{T}_{m} , and show that its centralizer in \mathcal{A}_{m} to be a split extension of \langle a \rangle^{*} by \mathcal{A}_s . We also describe important \mathbb{Z}_n [\mathcal{A}_s] submodules of \langle a\rangle^{*} .

Maximal Subgroups of the Group PSL(11, 2)

In this note, we will determine, up to the conjugacy, all the maximal subgroups of PSL(11, 2) by Aschbacher’s theorem.

Weakly $S$-semipermutable subgroups and $p$-nilpotency of groups

A subgroup H of a finite group G is said to be S -semipermutable in G if HG_{p}=G_{p}H for every Sylow subgroup G_{p} of G with (|H|,p)=1 . A subgroup H of G is said to be weakly S -semipermutable in G if there exists a normal subgroup T of G such that HT is S -permutable and H\cap T is S -semipermutable in G . In this paper we prove that for a finite group G , if some cyclic subgroups or maximal subgroups of G are weakly S -semipermutable in G , then G is p -nilpotent.

On σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-Solubility Criteria for Finite Groups

In this paper, the structure of finite groups in which maximal subgroups of some Sylow subgroups have a σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-soluble or σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-nilpotent supplement, where σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} is a partition of the set of all prime numbers, is investigated. Some solubility, σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-solubility and σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-nilpotency criteria leading to some significant improvements of earlier results are given.

Small G-varieties

AbstractAn affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group ${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the ${\mathbb {K}^{*}}$ -variety $X^U$ of fixed points under a maximal unipotent subgroup $U \subset G$ . Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient $X /\!\!/ G$ .If G is of type ${\mathsf {A}_n}$ ( $n\geq 2$ ), ${\mathsf {C}_{n}}$ , ${\mathsf {E}_{6}}$ , ${\mathsf {E}_{7}}$ , or ${\mathsf {E}_{8}}$ , we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If $n \geq 5$ , every smooth affine $\operatorname {\mathrm {SL}}_n$ -variety of dimension $< 2n-2$ is an $\operatorname {\mathrm {SL}}_n$ -vector bundle over the smooth quotient $X /\!\!/ \operatorname {\mathrm {SL}}_n$ , with fiber isomorphic to the natural representation or its dual.

Конечные группы с перестановочными строгими обобщенно максимальными подгруппами

The structure of finite groups in which any strictly 2-maximal subgroup permutes with an arbitrary strictly 3-maximal subgroup is described. It is shown that the class of groups with this property coincides with the class of groups in which any 2-maximal subgroup permutes with an arbitrary 3-maximal subgroup, and, as a consequence, such groups are solvable. As auxiliary results, we describe the structure of groups in which any strictly 2-maximal subgroup permutes with an arbitrary maximal subgroup. In particular, it is shown that the class of such groups coincides with the class of groups in which any 2-maximal subgroup commutes with all maximal subgroups, and, as a consequence, such groups are supersoluble.

Группы, в которых каждая строго 3-максимальная подгруппа перестановочна со всеми максимальными подгруппами

The description of the structure of finite non-nilpotent groups in which every strongly 3-maximal subgroup permutes with all maximal subgroups is the main result of this paper.In particular, we prove that the groups with the noted property are solvable and their structure is equivalent to the structure of groups in which every maximal subgroup permutes with all 3-maximal subgroups.The obtained results also prove the nilpotency of a finite group in which every strongly 3-maximal subgroup permutes with all maximal subgroupsif the number of prime divisors of the order of this group is not less than 4.

Finite groups whose maximal subgroups of even order are MSN-groups

Abstract A finite group G G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G G are subnormal in G G . In this article, we investigate the structure of finite groups G G such that G G is a non-MSN-group of even order in which every maximal subgroup of even order is an MSN-group. In addition, we determine the minimal simple groups all of whose second maximal subgroups are MSN-groups.