Sylow Subgroups Research Articles

On NH-embedded and SS-quasinormal subgroups of finite groups

Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in $G$ if $HG_p$ = $G_pH$ for any $G_p\in Syl_p(G)$ with $(|H|, p) = 1$, where $p$ is a prime number divisible $|G|$. Furthermore, $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H \cap T \leq H_{\overline{s}G}$, where $H_{\overline{s}G}$ is the largest $S$-semipermutable subgroup of $G$ contained in $H$, and $H$ is said to be $SS$-quasinormal in $G$ provided there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. In this paper, we obtain some criteria for $p$-nilpotency and Supersolvability of a finite group and extend some known results concerning $NH$-embedded and $SS$-quasinormal subgroups.

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

A note on Hall normally embedded subgroups of finite groups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called Hall normally embedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of the normal closure [Formula: see text]. In this paper, we fix a subgroup [Formula: see text] of Sylow subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that all subgroups [Formula: see text] of [Formula: see text] with [Formula: see text] are Hall normally embedded in [Formula: see text].

A note on Huppert's theorem and Chen's theorem

Let G be a finite group. We prove that every maximal subgroup of G has prime index if and only if every maximal subgroup of G that contains the normalizer of some Sylow subgroup has prime index, which implies that the hypothesis in Huppert's theorem and the hypothesis in Chen's theorem are actually equivalent. Moreover, we prove that the hypothesis in a theorem of Shao and Beltrán and the hypothesis in a theorem of Li et al. are also equivalent.

Periodic Groups with One Finite Nontrivial Sylow 2-Subgroup

Periodic Groups with One Finite Nontrivial Sylow 2-Subgroup

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.

Groups with three projective characters

Working over the field of complex numbers, the inequivalent irreducible projective representations of a finite group G with 2-cocycle α are considered. The greatest common divisor of the degrees of the α-characters of these representations is used to obtain information about the degrees of the irreducible αS -characters of a Sylow subgroup S of G. Suppose now that G has exactly three irreducible α-characters. Then the potential form that the three degrees can take is found if G has a nontrivial cyclic Sylow subgroup. However, the main result is to show that G is solvable when the degrees are of the form a, ra and ka with certain restrictions on r and k, most notably that r = 1 or k. This involves detailed analysis of the possible actions of G on the irreducible αN -characters of a normal subgroup N of G.

Periodic linear groups in which permutability is a transitive relation

A PT-group is a group in which the relation of being a permutable subgroup is transitive. The main aim of this paper is to show that a (homomorphic image of a) periodic linear group is a soluble PT-group if and only if each subgroup of a Sylow subgroup is permutable in the corresponding Sylow normalizer (see Theorem 4.7); for a fixed prime p, the latter condition is denoted by mathfrak {X}_p. In order to prove our main theorem, we need (i) to characterize (homomorphic images of) periodic linear groups that are PT-groups (see Sect. 2), (ii) to develop a fusion theory for locally finite groups (see Sect. 3), (iii) to carefully study (homomorphic images of) periodic linear groups with the property mathfrak {X}_p for a fixed prime p (see for instance Theorem 4.6). As a by-product we obtain (among other results) a characterization of (homomorphic images of) periodic linear mathfrak {X}_p-groups in terms of pronormality (see Theorem 4.11) that will allow us to show that, on some occasions, the property mathfrak {X}_p is inherited by 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

On finite groups in which some particular maximal invariant subgroups have indices a prime-power

Suppose that A and G are finite groups such that A acts coprimely on G by automorphisms, we first prove that if every maximal A-invariant subgroup of G that contains the normalizer of some A-invariant Sylow subgroup has index a prime-power and the projective special linear group is not a composition factor of G, then G is solvable. Moreover, we prove that if every non-nilpotent maximal A-invariant subgroup of G has index a prime-power and is not a composition factor of G, then G is solvable.

Characters, commutators and centers of Sylow subgroups

The character table of a finite group G G determines whether | P : P ′ | = p 2 |P:P’|=p^2 and whether | P : Z ( P ) | = p 2 |P:\mathbf {Z}(P)|=p^2 , where P P is a Sylow p p -subgroup of G G . To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.

Module braces: relations between the additive and the multiplicative groups

In this paper, we define a class of braces that we call module braces or R-braces, which are braces for which the additive group has also a module structure over a ring R, and for which the values of the gamma functions are automorphisms of R-modules. This class of braces has already been considered in the literature in the case where the ring R is a field; we generalise the definition to any ring R, reinterpreting it in terms of the so-called gamma function associated with the brace, and prove that this class of braces enjoys all the natural properties one can require. We exhibit explicit example of R-braces, and we study the splitting of a module braces in relation to the splitting of the ring R, generalising thereby Byott’s result on the splitting of a brace with nilpotent multiplicative group as a sum of its Sylow subgroups. The core of the paper is in the last two sections, in which, using methods from commutative algebra and number theory, we study the relations between the additive and the multiplicative groups of an R-brace showing that if a certain decomposition of the additive group is small (in some sense which depends on R), then the additive and the multiplicative groups have the same number of elements of each order. In some cases, this result considerably broadens the range of applications of the results already known on this issue.

Character Theory and Categorification

Over a hundred years after the work of Frobenius and Schur, the sheer enormity of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated. How does one decompose the tensor product of a pair of complex characters? Or the restriction of a complex character to a Sylow or wreath product subgroup? Can we understand the vanishing sets of complex characters? What about the asymptotic behaviour of complex characters? What are the dimensions of the modular characters? These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory. The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups.

Sylow Branching Coefficients and Hook Partitions

We give a description of the irreducible constituents of the restriction to Sylow 2-subgroups of irreducible characters of symmetric groups labelled by hook partitions.

Groups with 2-generated Sylow subgroups and their character tables

Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.

Characterizing subgroup perfect codes by 2-subgroups

A perfect code in a graph $$\Gamma $$ is a subset C of $$V(\Gamma )$$ such that no two vertices in C are adjacent and every vertex in $$V(\Gamma ){\setminus } C$$ is adjacent to exactly one vertex in C. Let G be a finite group and C a subset of G. Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of 2-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $$\textrm{PSL}(2,q)$$ is given.

On the inductive blockwise Alperin weight condition for type [formula omitted

In this paper we prove the blockwise Alperin weight conjecture for finite special linear and unitary groups, for finite groups with abelian Sylow 3-subgroups, and verify the inductive blockwise Alperin weight condition for certain cases of groups of type A. We also give a classification for the 2-blocks of special linear and unitary groups.

Orders of products of elements and nilpotency of terms in the lower central series and the derived series

In this paper we prove that if G is a finite group, then the k -th term of the lower central series is nilpotent if and only if for every \gamma_{k} -values x,y \in G with coprime orders, either \pi(o(x)o(y))\subseteq \pi(o(xy)) or o(x)o(y) \leq o(xy) . We obtain an analogous version for the derived series of finite solvable groups, but replacing \gamma_{k} -values by \delta_{k} -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.

On p′-degree ℓ-Brauer characters of ℓ-solvable groups

Let [Formula: see text] and [Formula: see text] be distinct primes, let [Formula: see text] be a finite [Formula: see text]-solvable group and let [Formula: see text]. Then [Formula: see text] has a unique irreducible [Formula: see text]-Brauer character of [Formula: see text]-degree lying over [Formula: see text] if and only if NG (P)/ P is an [Formula: see text]-group. This extends a result from [J., F.,Tent, Correspondences of Brauer characters and Sylow subgroup normalizers, J. Algebra 573 (2021) 436–450] to every odd prime [Formula: see text].

HALL SUBGROUPS AND -COCYCLE REGULARITY

AbstractLet H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$ -cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $ -regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $ -regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$ -regular element of a normal Hall subgroup H is $\alpha $ -regular in $G.$