Sylow Subgroups Research Articles

Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G, and H ( G ) is the set of all H -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G. As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ N G ( P ) is p-nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S-quasinormal in G, which means that H T permutes with every Sylow subgroup of G.

Finite groups with given sets of 𝔉-subnormal subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

Approximating Simple Locally Compact Groups by Their Dense Locally Compact Subgroups

Abstract The class $\mathscr{S}$ of totally disconnected locally compact (tdlc) groups that are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups, which studies the interaction between the compact open subgroups and the global structure, has emerged. In this article, we study the non-discrete tdlc groups $H$ that admit a continuous embedding with dense image into some $G\in \mathscr{S}$; that is, we consider the dense locally compact subgroups of groups $G\in \mathscr{S}$. We identify a class $\mathscr{R}$ of almost simple groups that properly contains $\mathscr{S}$ and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same properties previously obtained for $\mathscr{S}$ and establish various original results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$, notably concerning the structure of the local Sylow subgroups and the full automorphism group.

Permutability of the Sylow 2-subgroup with some biprimary subgroups

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L2(7) type, if p > 3, and additionally of L2(3f) type, f = 3a, a ≥ 1, if p = 3.

On m-S-supplemented subgroups of finite groups

Let G be a finite group and H a subgroup of G. We say that H is m-S-permutable in G, if for some modular subgroup A and S-permutable subgroup B of G; m-S-supplemented in G if there are an m-S-permutable subgroup S and a subgroup T of G such that G = HT and . In this article, we study finite groups with given systems of m-S-supplemented subgroups. In particular, we prove that if E is a normal subgroup of G such that for any noncyclic Sylow subgroup P of E all maximal subgroups of P or all cyclic subgroups of P of prime order and order 4 (in the case when P is a non-Abelian 2-group) are m-S-supplemented in G, then E is hypercyclically embedded in G.

Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

AbstractThe aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

The FSZ properties of sporadic simple groups

We investigate a possible connection between the [Formula: see text] properties of a group and its Sylow subgroups. We show that the simple groups [Formula: see text] and [Formula: see text], as well as all sporadic simple groups with order divisible by [Formula: see text] are not [Formula: see text], and that neither are their Sylow 5-subgroups. The groups [Formula: see text] and [Formula: see text] were previously established as non-[Formula: see text] by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be [Formula: see text]. We conclude by considering all perfect groups available through GAP with order at most [Formula: see text], and show they are non-[Formula: see text] if and only if their Sylow 5-subgroups are non-[Formula: see text].

A coprime action version of a solubility criterion of Deskins

Let A and G be finite groups of relatively prime orders and suppose that A acts on G via automorphisms. We demonstrate that if G has a maximal A-invariant subgroup M that is nilpotent and the Sylow 2-subgroup of M has class at most 2, then G is soluble. This result extends, in the context of coprime action, a solubility criterion given by W.E. Deskins.

Algorithm for calculation in Sylow 2-subgroups of alternating groups using the computer algebra system GAP

Algorithm for calculation in Sylow 2-subgroups of alternating groups using the computer algebra system GAP

On the absence of a normal nonabelian Sylow subgroup

Let G be a finite solvable group. We show that G does not have a normal nonabelian Sylow p-subgroup when its prime character degree graph satisfies a technical hypothesis.

A note on restriction of characters of alternating groups to Sylow subgroups

We restrict irreducible characters of alternating groups of degree divisible by p to their Sylow p-subgroups and study the number of linear constituents.

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

The expected number of elements to generate a finite group with $d$-generated Sylow subgroups

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

Sylow Subgroups and Fusion of Characters

Suppose that P is a Sylow p-subgroup of G. We study irreducible characters of P that are constant on G-fused classes.

The structure of finite groups with trait of non-normal subgroups

In this paper, we investigate the finite groups all of whose non-normal nilpotent subgroups are cyclic. We show that such groups are solvable with cyclic centers. If G is a non-supersolvable group, then G has only one non-cyclic Sylow subgroup which is either isomorphic to Q8 or is of type (q, q).

Finite groups with two supersoluble subgroups

Abstract Let G be a finite group. In this paper we obtain some sufficient conditions for the supersolubility of G with two supersoluble non-conjugate subgroups H and K of prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove that G is supersoluble in the following cases: one of the subgroups H or K is nilpotent; the derived subgroup G ′ {G^{\prime}} of G is nilpotent; | G : H | = q > r = | G : K | {|G:H|=q>r=|G:K|} and H is normal in G. Also the supersolubility of G with two non-conjugate maximal subgroups M and V is obtained in the following cases: all Sylow subgroups of M and of V are seminormal in G; all maximal subgroups of M and of V are seminormal in G.

On weakly s-semipermutable or ss-quasinormal subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup $$H_{ssG}$$ of G contained in H such that $$G=HT$$ and $$H\cap T\le H_{ssG}$$; H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that $$G=HB$$ and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying $$1<|D|<|P|$$ and study the structure of G under the assumption that every subgroup H of P with $$|H|=|D|$$ is either weakly s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified.

On the First Zassenhaus Conjecture and Direct Products

AbstractIn this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

Finite groups with some subgroups of Sylow subgroups weakly $\mathcal{H}$-embedded

Finite groups with some subgroups of Sylow subgroups weakly $\mathcal{H}$-embedded

Low Growth Equational Complexity

AbstractThe equational complexity function $\beta \nu \,:\,{\open N} \to {\open N}$ of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting $O(\log_{2}^{3}(n))$ growth among varieties of semilattice-ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.