P-subgroup Research Articles

Modular invariants of finite gluing groups

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow p-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic p-group acting on an elementary abelian p-group.

On finite p-groups of supersoluble type

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

Even character degrees and normal Sylow 2-subgroups

Abstract The Ito–Michler theorem on character degrees states that if a prime p does not divide the degree of any irreducible character of a finite group G, then G has a normal Sylow p-subgroup. We give some strengthened versions of this result for p = 2 {p=2} by considering linear characters and those of even degree.

On the number of p-elements in a finite group

In this paper, we study the ratio between the number of p-elements and the order of a Sylow p-subgroup of a finite group G. As well known, this ratio is a positive integer and we conjecture that, for every group G, it is at least the $$(1-\frac{1}{p})$$ -th power of the number of Sylow p-subgroups of G. We prove this conjecture if G is p-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.

On nilpotency of higher commutator subgroups of a finite soluble group

Let G be a finite soluble group and $$G^{(k)}$$ the kth term of the derived series of G. We prove that $$G^{(k)}$$ is nilpotent if and only if $$|ab|=|a||b|$$ for any $$\delta _k$$ -values $$a,b\in G$$ of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then $$P\cap G^{(k)}$$ is generated by $$\delta _k$$ -values contained in P (Lemma 2.5). This is related to the so-called focal subgroup theorem.

Galois action on the principal block and cyclic Sylow subgroups

We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.

A note on bounds of the p-length of a p-soluble group

Suppose that the finite group $$G=AB$$ is a mutually permutable product of two p-soluble subgroups A and B. By using the Wielandt lenghts of Sylow p-subgroups of A and B, we present a new bound of the p-length of G. Some known results are improved.

On principal indecomposable degrees and Sylow subgroups

We conjectured in Malle and Navarro (J Algebra 370:402–406, 2012) that a Sylow p-subgroup P of a finite group G is normal if and only if whenever p does not divide the multiplicity of $$\chi \in {{\text {Irr}}}(G)$$ in the permutation character $$(1_P)^G$$ , then p does not divide the degree $$\chi (1)$$ . In this note, we prove an analogue of this for p-Brauer characters.

A solvability criterion for finite groups related to the number of Sylow subgroups

Let G be a finite group and let be the set of primes dividing the order of G. For each the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp + 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most Sylow p-subgroups for each In particular, we show that each finite group G containing at most Sylow p-subgroups for each is solvable.

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

AbstractFor each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$, $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$.

Finite groups in which the sylowizers of each non-trivial primary subgroups are conjugate

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we investigate the structure of the group G in which the sylowizers of each non-trivial primary subgroups in G are conjugate and give the non-solvable chief factors of G.

Galois structure of the holomorphic differentials of curves

Let X be a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Suppose G is a finite group acting faithfully on X such that G has non-trivial cyclic Sylow p-subgroups. We show that the decomposition of the space of holomorphic differentials of X into a direct sum of indecomposable k[G]-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of X that are ramified in the cover X⟶X/G. We apply our method to determine the PSL(2,Fℓ)-module structure of the space of holomorphic differentials of the reduction of the modular curve X(ℓ) modulo p when p and ℓ are distinct odd primes and the action of PSL(2,Fℓ) on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing p between modular forms arising from isotypic components with respect to the action of PSL(2,Fℓ) on X(ℓ).

Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case

Let p,q be distinct primes, with p>2.We classify the Hopf-Galois structures on Galois extensions of degree p2q, such that the Sylow p-subgroups of the Galois group are cyclic.This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘).Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G.Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅g−1). These functions are in one-to-one correspondence with the regular subgroups of the holomorph of G, and are characterised by the functional equation γ(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.

Nearly s-embedded subgroups and the p-nilpotency of finite groups

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that and where HsG is the intersection of all s-permutable subgroups of G containing H. In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow p-subgroups on the p-nilpotency of finite groups. A number of known results are improved and extended.

On heights of characters of finite groups

In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z(G) isomorphic to A6,A7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group, as well as for all blocks of finite general linear or unitary groups. For the proof of the case with symmetric groups, it relates to an open question of Olsson on the existence of t-core partitions, where t≥3 is an integer. As a byproduct, our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the Isaacs-Moretó-Navarro-Tiep Conjecture, claiming that the number of distinct irreducible character degrees of a Sylow p-subgroup P of an arbitrary finite group G is at most one more than the number of irreducible character degrees of G that are multiples of p.

The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of A 2 k is constructed. It is shown that ( S y l 2 A 2 k ) 2 = S y l 2 ′ A 2 k , k > 2 . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups C p i , p i ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) of A 2 k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group A 2 k , where k > 2 , the permutation group S 2 k , as well as Sylow p-subgroups of S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B ≀ C n is investigated. An upper bound of the commutator width of permutational wreath product B ≀ C n for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.

English

For a finite group G and a fixed Sylow p-subgroup P of G, Ballester-Bolinches and Guo proved in 2000 that G is p-nilpotent if every element of P ∩ G′ with order p lies in the center of NG(P) and when p = 2, either every element of P ∩ G′ with order 4 lies in the center of NG(P) or P is quaternion-free and NG(P) is 2-nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p-nilpotent if every element of P with order p is weakly pronormal in G and when p = 2, every element of P with order 4 is also weakly pronormal in G. These results generalized famous Ito’s Lemma. We are motivated to generalize Ballester-Bolinches and Guo’s Theorem and Asaad’s Theorem. It is proved that if p is the smallest prime dividing the order of a group G and P, a Sylow p-subgroup of G, then G is p-nilpotent if G is S4-free and every subgroup of order p in P ∩ Px ∩ $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is weakly pronormal in NG(P) for all x ∈ GNG(P), and when p = 2, P is quaternion-free, where $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is the p-nilpotent residual of G.

5G Massive Multiple Input and Multiple Output System with Maximum Spectral Performance

Massive MIMO is the presently maximum compelling sub-6 GHz physical-layer era for destiny wireless access. Excellent spectral overall performance finished through manner of spatial multiplexing of many terminals inside the equal time-frequency resource. The 5G structures are characterized through excessive transmission records prices, 1Gbps and above, so large bandwidth transmission is expected. The most vital objectives within the 5G wireless systems design is to deal with the excessive inter photograph interference (ISI) as a consequence of the high statistics fees, and using the accessible spectral bandwidth in resourceful way.MC-CDMA is categorized in to two methods that is ODFM and CDMA. Efficient useful resource allocation is the principle trouble within the development of fourth generation cellular communication systems. For utilizing the internet and multimedia a maximum data fee is preferred, so in this paper the general performance of MC-CDMA systems the usage of Sylow theorem for grouping this is executed that's a spectrum allocation technique is presented. This paper particularly analyzes the presence of Additive white Gaussian Noise (AWGN) in MC-CDMA utilizing QPSK for special variety of subcarrier, exclusive wide sort of customers with the help of MATLAB tool. This paper shows the reduction in BER and power allocation among the MC-CDMA and massive-MIMO.

On zeros of irreducible characters lying in a normal subgroup

Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that $$\chi (g)\ne 0$$ for all irreducible characters $$\chi$$ of G. Such an element is said to be non-vanishing in G. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if $$N=G$$ , then new contributions are obtained.

On relative projectivity of lattices in Auslander-Reiten components for group rings

Let G be a finite group, Q a p-subgroup of G and P a Sylow p-subgroup of Let be a complete discrete valuation ring of characteristic zero with residue class field of characteristic p > 0. Suppose that the group ring is of infinite representation type and is sufficiently large. Let be a stable Auslander-Reiten component containing a Scott -lattice S(Q) with vertex Q. Then all the -lattices in have P as their vertices. Also, we show that there exists an indecomposable -lattice L with vertex P such that a kG-module is Q-projective.Communicated by J. Zhang