Abelian Sylow Subgroups Research Articles

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups

Profinite groups with abelian Sylow subgroups

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

Influence of complemented subgroups on the structure of finite groups

P‎. ‎Hall proved that a finite group $G$ is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of $G$ is complemented in $G$‎. ‎He called such groups complemented‎. ‎A‎. ‎Ballester-Bolinches and X‎. ‎Guo‎ ‎established the structure of minimal non-complemented groups‎. ‎We give the classification of finite non-soluble groups‎ ‎all of whose second maximal subgroups are complemented groups‎. ‎We also prove that every finite group with less than‎ ‎$21$ non-complemented non-minimal ${2,3,5}$-subgroups is soluble‎.

О перестановочных строго 2-максимальных и строго 3-максимальных подгруппах

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

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.

Every finite abelian group is a subgroup of the additive group of a finite simple left brace

Left braces, introduced by Rump, have turned out to provide an important tool in the study of set-theoretic solutions of the quantum Yang–Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace (B,+,⋅) is a structure determined by two group structures on a set B: an abelian group (B,+) and a group (B,⋅), satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group A is a subgroup of the additive group of a finite simple left brace B with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.

The Influence of Conjugacy Class Sizes on the Structure of Finite Groups

Let $G$ be a group. The question of how certain arithmetical conditions on the sizes of the conjugacy classes of $G$ influence the group structure has been studied by many authors. In this paper, we investigate the influence of conjugacy class sizes of primary and biprimary elements on the structure of $G$. A criterion for a group to have abelian Sylow subgroups is given and some well-known results on Baer-groups are generalized.

On automorphisms of irreducible linear groups with an Abelian Sylow 2-subgroup

On automorphisms of irreducible linear groups with an Abelian Sylow 2-subgroup

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.

Abelian Sylow subgroups in a finite group, II

Let p be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p, and, if p∈{3,5}, the degree of every irreducible character in the principal p-block of G is coprime to p. This gives a complete solution to a problem posed by R. Brauer in 1963.

Constructions for Terraces and R-Sequencings, Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Every abelian group of even order with a noncyclic Sylow 2-subgroup is known to be R-sequenceable except possibly when the Sylow 2-subgroup has order 8. We construct an R-sequencing for many groups with elementary abelian Sylow 2-subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2-groups have terraces. For odd orders it is known that abelian groups are R-sequenceable except possibly those with noncyclic Sylow 3-subgroups. We show how the theory of narcissistic terraces can be exploited to find R-sequencings for many such groups, including infinitely many groups with each possible of Sylow 3-subgroup type of exponent at most 312 and all groups whose Sylow 3-subgroups are of the form or .

On the inductive Alperin–McKay and Alperin weight conjecture for groups with abelian Sylow subgroups

We study the inductive Alperin–McKay conjecture, the inductive Isaacs–Navarro refinement and the inductive blockwise Alperin weight conjecture for groups of Lie type in the generic case of abelian Sylow ℓ-subgroups. We also show that the alternating groups, the Suzuki groups and the Ree groups satisfy the inductive condition necessary for Späthʼs reduction of the blockwise Alperin weight conjecture to the case of simple groups.

Abelian Sylow subgroups in a finite group

Let p≠3,5 be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p. This gives a solution to a problem posed by R. Brauer in 1956 (for p≠3,5).

On a Fitting length conjecture without the coprimeness condition

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order pnq, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.

The p-local ranks of finite simple groups with abelian Sylow p-subgroups

As a continuation of previous work, in this paper, we mainly study the finite simple groups which have abelian Sylow subgroups in term of p-local rank, especially a group theoretic characterization will be given.

On torsion subgroups in integral group rings of finite groups

The object of this article are torsion subgroups of the normalized unit group V ( Z G ) of the integral group ring Z G of a finite group G. For specific subgroups W we study the Gruenberg–Kegel graph Π ( W ) . It is shown that the central elements of an isolated subgroup U of a group basis H of Z G are the normalized units of its centralizer ring C Z G ( U ) . Moreover Π ( N V ( Z G ) ( U ) ) = Π ( N H ( U ) ) . If G has elementary abelian Sylow 2-subgroups of order at most 8 each finite 2-subgroup of V ( Z G ) is rationally conjugate to a subgroup of G. Finally torsion subgroups of V ( Z G ) in the case when G is a minimal simple group are considered. It follows that if G is a simple group which admits a non-trivial partition for each prime p the p-rank of a torsion subgroup of V ( Z G ) is bounded by that one of G.

Simple modules for groups with abelian Sylow 2-subgroups are algebraic

An algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if G is a group with abelian Sylow 2-subgroups and K is a field of characteristic 2, then every simple KG-module is algebraic.

Reduction of [formula omitted] for finite groups with normal abelian Sylow 2-subgroup

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell’s unitary nilpotent groups UNil ∗ ( Z [ F ] ; Z [ F ] , Z [ F ] ) have an induced isomorphism to the quotient of UNil ∗ ( Z [ S ] ; Z [ S ] , Z [ S ] ) by the action of the group F / S . In particular, any finite group F of odd order has the same UNil -groups as the trivial group. The broader scope is the study of the L -theory of virtually cyclic groups, based on the Farrell–Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2-group and when S is a special 2-group.

Cyclical subnormal separation in A-groups

Three main results, concerning ,4-groups in respect of cyclical subnormal separation as defined in [4], are presented. It is shown in theorem A that any /1-group that, is generated by elements of prime order and satisfying the cyclical subnormal separation condition is metabelian. The two other main results give necessary and sufficient conditions for r4-groups, that are split extensions of certain abclian p-groups by a metabelian p' group, to satisfy the cyclical subnormal separation condition. There is also a result which shows that .4-groups with elementary abelian Sylow subgroups are cyclically separated as defined in [3]. 1-. Introduction A group G is called a CSn group if for any given cyclic subgroup B '„ group if and only if every element of // of prime order is in the Fitting subgroup of H. Theorem C Suppose that G = W » H is an /1-group where W is a minimal normal subgroup of G and // is a metabelian /)'-subgroup acting faithfully on W. Then G is a CSn group if and only if every elementof H\F(H) acts fixed-point-freely on 14-', where F(!l) is the Kitting subgroup of H. We also come across this result, Lemma (3.3) which concerns CS groups. We find it appropriate to bring it here because the CSpaper introduced in [3] was the motivating factor for the appearance of the CSn groups. There was the general feeling that /l-groups may be C.^-groups. Lemma (3.3) gave one positive result. But as it turns out we have an example of an A group which is even a CS7I group but not a CS-group. Section 2 deals with general results that are of interest to us including an example of a rion metabelian ,4-group in CSn. In fact as it has been pointed out in the concluding remarks ,i slight modification of this example gives an example of an /1-group in CSn which i> not a CS group. MIRAMARE TRIESTE December 1995 'Permanent address: Department of Mathematics, Ahrnadu Bello University, Samaru, Zaria, Kaduna State, Nigeria. 2 Preliminaries 2.1 Theorem A established that any A-group in CSn which is generated by elements of prime order must be metabelian. We begin our discussions with an example to show, first, that there are non metabelian /l-groups in CSn and secondly that the hypothesis, in the theorem, of the generating elements to be of prime order is necessary. We note that metabelian groups had already been shown to belong to the class of CSn groups [(3.5) of [4]]2.2 Example (2.2) Suppose that p, q and r are three distinct primes and (i) is a cyclic group of order r. Let k be the order of q mod r, i.e., r ^ j and r f IJ' — 1 for ( '') = 1Tliis implies that C'JX'O — 1. Clearly, O is an .4 group. To see why it is a f-'.S',, group we first observe tiiat C = L » H where II = M{.r) is a ;/-metabelian group, by [(3.!)) [4]] // G Ci>'n. Nest, any (; eleinent in // is conjugate to some m € M arid by n) denotes the smallest subnormal subgroup of // containing in. Note that. (m) = (m) since A/ is abelian and normal in //. Also, any r element of // is conjugate to an eleiLient in (x). But For 1 / ;/ 6 (•'')-,!> = •>' f° soim; v £ {1,2. . , . , r I [. If i; — r we have seen that f.'r,(x) = I. If » ^ r then y = :(:' and Hence by theorem A of [4] G is a CSn group. The following special case gives an idea of what G looks like. Let r = 2,q = 3 and k — 1 then M = (m) and (,r) are cyclic of orders !J and 1 respectively. Also M(.r> = {m,.r|m = J= l,m* = m'} H

Eigenvalues of Cartan matrices of principal 2-blocks with abelian defect groups

Let G be a finite group with an abelian Sylow 2-subgroup P. Let C B be the Cartan matrix of the principal 2-block B of G. We show that the Frobenius–Perron eigenvalue ρ ( B ) of C B is a rational integer if and only if B and its Brauer correspondent block b of N G ( P ) are Morita equivalent by using a classification of finite simple groups with an abelian Sylow 2-subgroup. In this case, we can take the Brauer character table Φ b of b as a unimodular eigenvector matrix U B of C B over a complete discrete valuation ring R.