Sylow Subgroups Research Articles

The Number of Subgroup Chains of Finite Nilpotent Groups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

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.

π-Quasinormally embedded subgroups and p-nilpotency of finite groups

Let P be a nontrivial p-group. A chain of subgroups is called a maximal chain of P provided that In this article, we determine the structure of finite groups having π-quasinormally embedded subgroups of a maximal chain of Sylow subgroups. We obtain new characterizations of finite p-nilpotent and supersolvable groups.

Finite groups with seminormal or abnormal Sylow subgroups

‎Let $G$ be a finite group in which every Sylow subgroup‎ ‎is seminormal or abnormal‎. ‎We prove that $G$ has a Sylow tower‎. ‎We establish that if a group has a maximal subgroup ‎‎‎‎with Sylow subgroups under the same conditions‎, ‎then this group is soluble‎.

Generation of finite groups with cyclic Sylow subgroups

Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

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.

The essential 2-rank of general linear and unitary groups

Let G be a general linear or unitary group over a finite field of odd characteristic and let D≤G be a Sylow 2-subgroup. In this paper, we classify essential subgroups of the Frobenius category FD(G) and determine the number of G-conjugacy classes of these subgroups, that is, the essential 2-rank of FD(G).

On finite groups with SS-supplemented subgroups

Let G be a finite group. A subgroup H of G is called S-permutable in G if H permutes with all Sylow subgroups of G. A subgroup H of G is called SS-supplemented in G if there exists a subgroup K of G such that HK = G and is S-permutable in K. In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of a Sylow subgroup of G are SS-supplemented in G, and obtain a group G is supersolvable if all cyclic subgroups of G of prime power order are SS-supplemented in G.

On commutator subgroups of Sylow 2-subgroups of the alternating group, and the commutator width in wreath products

We find the size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group and investigate the structure of the commutator subgroup of Sylow 2-subgroups of the alternating group $${A_{{2^{k}}}}$$ . This work continues previous investigations of the author (Skuratovskii in Cybern Syst Anal 45(1):25–37, 2009; ROMAI J 13(1):117–139, 2017), where minimal generating sets of Sylow 2-subgroups of alternating groups were found.

On the lattice of subquandles of a Takasaki quandle

Let A be an abelian group. If for any we define , then is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by . Indeed, we characterize the subquandles of T(A) and show that is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but is isomorphic to

On the residual of a factorized group with widely supersoluble factors

Let be the set of all primes. A subgroup H of G is called -subnormal in G, if either H = G, or there exists a chain Recall that G is -supersoluble, if every Sylow subgroup of G is -subnormal in G. The structure of -supersoluble residual of G = AB with -subnormal -supersoluble subgroups A and B is obtained.

On the number of Sylow subgroups in finite simple groups

Denote by [Formula: see text] the number of Sylow [Formula: see text]-subgroups of [Formula: see text]. For every subgroup [Formula: see text] of [Formula: see text], it is easy to see that [Formula: see text], but [Formula: see text] does not divide [Formula: see text] in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group [Formula: see text] satisfies DivSyl(p) if [Formula: see text] divides [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text]. In this paper, we show that “almost for every” finite simple group [Formula: see text], there exists a prime [Formula: see text] such that [Formula: see text] does not satisfy DivSyl(p).

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.

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.

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.

Lamplighter groups, bireversible automata, and rational series over finite rings

We realize lamplighter groups A\wr \mathbb Z , with A a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize A\wr \mathbb Z as a bireversible automaton group if and only if the 2-Sylow subgroup of A has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.

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 $\log \log {n}/\log \log \log {n}$.

Primary subgroups and the structure of finite groups

Let G be a group and The permutizer of H in G is H is said to be strongly permuteral in G if whenever Moreover, let is a primary element of H is said to be strongly -permuteral in G if whenever In this article, we study the structure of a group G in which every Sylow subgroup and its maximal subgroups are strongly -permuteral or abnormal and the structure of G with self-normalizing or strongly permuteral Sylow subgroups.

On the supersolubility of a group with semisubnormal factors

Abstract A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that G = A ⁢ B {G=AB} and AX is a subgroup of G for every subgroup X of B. We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semisubnormal in G if A is subnormal in G or seminormal in G. A group G = A ⁢ B {G=AB} with semisubnormal supersoluble subgroups A and B is studied. The equality G 𝔘 = ( G ′ ) 𝔑 {G^{\mathfrak{U}}=(G^{\prime})^{\mathfrak{N}}} is established; moreover, if the indices of subgroups A and B in G are relatively prime, then G 𝔘 = G 𝔑 2 {G^{\mathfrak{U}}=G^{\mathfrak{N}^{2}}} . Here 𝔑 {\mathfrak{N}} , 𝔘 {\mathfrak{U}} and 𝔑 2 {\mathfrak{N}^{2}} are the formations of all nilpotent, supersoluble and metanilpotent groups, respectively; H 𝔛 {H^{\mathfrak{X}}} is the 𝔛 {\mathfrak{X}} -residual of H. Also we prove the supersolubility of G = A ⁢ B {G=AB} when all Sylow subgroups of A and of B are semisubnormal in G.

Correction to: Normalizers of Sylow Subgroups in Finite Linear and Unitary Groups

Correction to: Normalizers of Sylow Subgroups in Finite Linear and Unitary Groups