Large Abelian Subgroups Research Articles

The character degree ratio and large abelian subgroups

Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. M. Isaacs defined the character degree ratio of G to be rat(G)=b(G)/m(G). He proved that for nonabelian solvable groups the derived length is bounded by a logarithmic function of the character degree ratio. In a similar spirit, results that restrict the structure of nonsolvable groups in terms of the character degree ratio were later proved by J. P. Cossey, M. Lewis and H. N. Nguyen. In this note we show that, with some rather precisely determined exceptions, G has an abelian subgroup of index bounded by a polynomial function in rat(G). As a consequence we get stronger (in most situations) versions of most of the previously known results on the character degree ratio.

The Highest Dimension of Commutative Subalgebras in Chevalley Algebras

Let L (K) denotes a Chevalley algebra with the root system over a field K. In 1945 A. I. Mal’cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra. Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups. The main result of this article allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra L (K) over an arbitrary field

Cryptanalysis of cryptosystems based on general linear group

Advances in quantum computers threaten to break public key cryptosystems such as RSA, ECC, and EIGamal on the hardness of factoring or taking a discrete logarithm, while no quantum algorithms are found to solve certain mathematical problems on non-commutative algebraic structures until now. In this background, Majid Khan et al. proposed two novel public-key encryption schemes based on large abelian subgroup of general linear group over a residue ring. In this paper we show that the two schemes are not secure. We present that they are vulnerable to a structural attack and that, it only requires polynomial time complexity to retrieve the message from associated public keys respectively. Then we conduct a detailed analysis on attack methods and show corresponding algorithmic description and efficiency analysis respectively. After that, we propose an improvement assisted to enhance Majid Khan's scheme. In addition, we discuss possible lines of future work.

A Novel Cryptosystem Based on General Linear Group

In this article, we have developed a novel public-key cryptosystem that uses large abelian subgroup of general linear group over residue ring. The merit of this proposed public key cryptosystem is that we can select session key in abelian subgroup of general linear group which reduces exponentiations and performed encryption effectively in a simple way. Our algorithm doesn't use matrix modular exponentiation which leads us to problem in implementing. The aim of this article is to decrease the number of these exponentiations and consequently to accelerate the execution of encryption algorithm. A discussion about the security of built modifications made in the article shows that the level of security is high enough for an appropriate choice of parameters of the cryptosystems.

The normal structure of the unipotent subgroup in Lie type groups and its extremal subgroups

We study the normal structure of the unipotent radical U of a Borel subgroup in a Lie type group over a field K. Thus, all maximal Abelian normal subgroups in U are described. This gives a new solution of C. Parker and P. Rowley’s problem about extremal subgroups in U and the description in finite groups U of the large normal (and, as proved, also normal large) Abelian subgroups.

Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type

We describe all maximal abelian normal subgroups in the unipotent radical U of a Borel subgroup in a group of Lie type G over a field K. This gives a new description of the extremal subgroups in U which were studied by C. Parker and P. Rowley. For a finite field K, we prove that either each large abelian subgroup in U is G-conjugate to a normal subgroup in U or G is of certain exceptional Lie type.

On conjugacy in a Chevalley group of large Abelian subgroups of the unipotent subgroup

Let U be the unipotent subgroup of a Chevalley group over a finite field. The well-known problem about describing the set of “large” (of maximal order) Abelian subgroups in U of exceptional type is investigated. The description of normal large Abelian subgroups in U was established earlier. It is proved that each large Abelian subgroup from U is conjugate in the Chevalley group of type F 4 over a finite field of characteristic not equal to 2 to a normal subgroup in U. It is shown that for the groups U of type G 2 and 3 D 4 a similar conclusion is not true.

Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups

The description of the automorphisms of a unipotent subgroup U of a Chevalley group over a field K known earlier for charK ≠ 2, 3 (Gibbs, 1970) was completed in 1990 together with the solution of problem (1.5) from A.S. Kondrat’ev’s survey (Usp. Mat. Nauk, 1986). In the present paper, Aut U is described for the case of finitary Chevalley groups. For a Chevalley group of classical type over a finite field, it is proved that any large Abelian subgroup from U is conjugate to a normal subgroup in U. It is shown that this is not so in the general case; therefore, problem (1.6) from Kondrat’ev’s survey about large Abelian subgroups in U is reduced to listing the exceptions. The orders of these subgroups and large Abelian normal subgroups in U were listed earlier.

Centrally large subgroups of finite p-groups

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if | A | ⩾ | A ∗ | for every abelian subgroup A ∗ of S. We say that a subgroup Q of S is a centrally large subgroup, or CL-subgroup, of S if | Q | ⋅ | Z ( Q ) | ⩾ | Q ∗ | ⋅ | Z ( Q ∗ ) | for every subgroup Q ∗ of S. The study of large abelian subgroups and variations on them began in 1964 with Thompson's second normal p-complement theorem [J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964) 43–46]. Centrally large subgroups possess some similar properties. In 1989, A. Chermak and A. Delgado [A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989) 907–914] studied several families of subgroups that include centrally large subgroups as a special case. In this paper, we extend their work to prove some further properties of centrally large subgroups. The proof uses an analogue for finite p-groups of an application of Borel's Fixed Point Theorem for algebraic groups.

Compact groups with large abelian subgroups

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Large Abelian Subgroups of Groups of Prime Exponent

Abelian subgroups, particularly ‘‘large’’ abelian subgroups of p-groups, have played an important role in finite group theory. Let p be a prime and let P be a finite p-group. Suppose A is an elementary abelian subgroup of some order p in P. We ask whether P has a normal elementary abelian subgroup of order p. In general, the answer is no, for example, if n 2 and P is the dihedral group of order 16. More generally, Alperin has constructed a counterexample for each prime p Hup, p. 349 . However, there are situations in which the answer must be yes, e.g., AG, Theorems A and 5.8 if p is odd and p 4n 7, if P has nilpotence class at most p, Ž p . or if p 3 and P has exponent p x 1 for every x P . Since Alperin’s counterexamples have exponent greater than p, this leaves the question open if p 5 and P has exponent p. The purpose of this paper is to give counterexamples in these cases. Ž . As usual, define the rank m A of a finite abelian group to be the Ž . minimal number of generators of A, and the rank r P of a finite p-group P to be the maximum of the ranks of the abelian subgroups of P. We prove:

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for 2F4(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F4(q), E6(q), E7(q), E8(q), and 2E6(q2)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F4(q), E6(q), E7(q), E8(q), and 2E6(q2)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for 2F4(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.

Large Abelian Subgroups of Finitep-Groups

Ž . Let p be a prime and let S be a finite p-group. Define d S to be the < < Ž . maximum of A as A ranges over the abelian subgroups of S, and A S to < < Ž . Ž . be the set of all abelian subgroups A for which A s d S . Let J S be Ž . the subgroup of S generated by A S . w x Ž . In 1964, Thompson T2 introduced a subgroup similar to J S and used it to obtain an improved version of his first normal p-complement theorem w x Ž . T1 . Since then, several variants of J S have been used to obtain related results. Most of their proofs have required a further result of Thompson, w x the Replacement Theorem Gor, p. 273; HB, III, p. 21 . Ž . Suppose A g A S , B is an abelian subgroup of S normalized by A, and Ž . B does not normalize A. Then there exists A* g A S such that

A Large Nilpotent Group without Large Abelian Subgroups

A Large Nilpotent Group without Large Abelian Subgroups

On the Structure of Twisted Group C ∗ -Algebras

We first give general structural results for the twisted group algebras C*(G, σ) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, σ) as the sections of a C*-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ is a discrete subgroup of a solvable Lie group G, the K-groups K * (C*(Γ, σ)) are isomorphic to certain twisted K-groups K*(G/Γ, δ(σ)) of the homogeneous space G/Γ, and we discuss how the twisting class δ(σ) ∈ H 3 (G/Γ, Z) depends on the cocycle σ

On the structure of twisted group 𝐶*-algebras

We first give general structural results for the twisted group algebras C ∗ ( G , σ ) {C^{\ast } }(G,\sigma ) of a locally compact group G G with large abelian subgroups. In particular, we use a theorem of Williams to realise C ∗ ( G , σ ) {C^{\ast }}(G,\sigma ) as the sections of a C ∗ {C^{\ast }} -bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ \Gamma is a discrete subgroup of a solvable Lie group G G , the K K -groups K ∗ ( C ∗ ( Γ , σ ) ) {K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma )) are isomorphic to certain twisted K K -groups K ∗ ( G / Γ , δ ( σ ) ) {K^{\ast } }(G/\Gamma ,\delta (\sigma )) of the homogeneous space G / Γ G/\Gamma , and we discuss how the twisting class δ ( σ ) ∈ H 3 ( G / Γ , Z ) \delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb {Z}) depends on the cocycle σ \sigma . For many particular groups, such as Z n {\mathbb {Z}^n} or the integer Heisenberg group, δ ( σ ) \delta (\sigma ) always vanishes, so that K ∗ ( C ∗ ( Γ , σ ) ) {K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma )) is independent of σ \sigma , but a detailed analysis of examples of the form Z n ⋊ Z {\mathbb {Z}^n} \rtimes \mathbb {Z} shows this is not in general the case.

Large abelian subgroups of Chevalley groups

AbstractFor any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.

Large abelian subgroups of some infinite froups, II

Large abelian subgroups of some infinite froups, II

Large abelian subgroups of some infinite groups

Large abelian subgroups of some infinite groups

Large Abelian Subgroups of p-Groups

Large Abelian Subgroups of p-Groups