Normal Sylow Research Articles

Skew-morphisms of elementary abelian 𝑝-groups

Abstract A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

P-Singular characters and normal Sylow p-subgroups

Let [Formula: see text] be a finite group and [Formula: see text] a prime. We denote by [Formula: see text] the set of irreducible complex characters of [Formula: see text] whose degrees are linear or divisible by [Formula: see text], and we write [Formula: see text] to denote the ratio of the sum of squares of irreducible character degrees in [Formula: see text] to the sum of irreducible character degrees in [Formula: see text]. The Itô–Michler Theorem on character degrees states that [Formula: see text] if and only if [Formula: see text] has a normal abelian Sylow [Formula: see text]-subgroup. We generalize this theorem as follows: if [Formula: see text], then [Formula: see text] has a normal Sylow [Formula: see text]-subgroup.

Finite groups with many p-regular conjugacy classes

Let G be a finite group and let p be a prime. In this paper, we study the structure of finite groups with a large number of p-regular conjugacy classes or, equivalently, a large number of irreducible p-modular representations. We prove sharp lower bounds for this number in terms of p and the p′-part of the order of G which ensure that G is p-solvable. A bound for the p-length is obtained which is sharp for odd primes p. We also prove a new best possible criterion for the existence of a normal Sylow p-subgroup in terms of these quantities.

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.

The inductive McKay–Navarro condition for the Suzuki and Ree groups and for groups with non-generic Sylow normalizers

We verify the inductive McKay–Navarro condition for the Suzuki and Ree groups for all primes as well as for ℓ=3 and the groups PSL3(q) with q≡4,7mod9, PSU3(q) with q≡2,5mod9, and G2(q) with q≡2,4,5,7mod9. On the way, we show that there exists a Galois-equivariant Jordan decomposition for the irreducible characters of the Suzuki and Ree 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.

Orientably-regular p-maps and regular p-maps

A map is called a p-map if it has a prime p-power vertices. An orientably-regular (resp. A regular) p-map is called solvable if the group G+ of all orientation-preserving automorphisms (resp. the group G of automorphisms) is solvable; and called normal if G+ (resp. G) contains the normal Sylow p-subgroup.In this paper, it will be proved that both orientably-regular p-maps and regular p-maps are solvable and except for few cases that p∈{2,3}, they are normal. Moreover, nonnormal p-maps will be characterized and some properties and constructions of normal p-maps will be given.

A class of orientably‐regular embeddings of graphs of order [formula omitted

Let G be the automorphism group of an orientably regular embedding of a connected simple graph of order p3, where p is a prime. It was shown in [19] that except for the case when p=2 and G=S4, G contains a normal Sylow p-subgroup P of order pk where k∈{3,4,5}, while such embeddings with k=5 were determined. In this paper, a further classification is achieved for all such embeddings with k=3.

Some identities on twisted group algebras

Let G be a group and F be a field of characteristic . In this paper we provide some necessary and sufficient conditions for a twisted group algebra to satisfies a non-matrix identity. This allows us to show that a crossed product is Lie nilpotent if and only if σ is trivial, G is nilpotent and p-abelian, G has a unique normal Sylow p-subgroup P and for some central -subgroup Q and is stably untwisted. Also, is Lie nilpotent if and only if is commutative. Some generalized group identities on the unit group of are also investigated.

Vanishing elements and Sylow subgroups

Let G be a finite group and p be a prime. A vanishing conjugacy class of G is a conjugacy class of G which consists of vanishing elements of G. A p-singular vanishing conjugacy class is a vanishing conjugacy class with elements of order divisible by p. Dolfi, Pacifici, Sanus and Spiga proved that if a group has no p-singular vanishing elements, then it has a normal Sylow p-subgroup. In this note we investigate the structure of a Sylow p-subgroup P of a group G with exactly one p-singular vanishing conjugacy class. In particular, we show that if p > 3, then P′ is a subnormal subgroup of G. We also prove that |P/ O p (G)| ≤ p or G has a composition factor isomorphic to PSL2(q), where q = pf , f ≥ 2 and O p (G) denotes the largest normal p-subgroup of G.

Finite Groups with Subnormal Residuals of Sylow Normalizers

Finite Groups with Subnormal Residuals of Sylow Normalizers

Sylow normalizers and irreducible characters with small cyclotomic field of values

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.

On ℳ-supplemented subgroups

Abstract Let 𝐺 be a finite group and p k p^{k} a prime power dividing | G | \lvert G\rvert . A subgroup 𝐻 of 𝐺 is said to be ℳ-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H ⁢ K G=HK and H i ⁢ K < G H_{i}K<G for every maximal subgroup H i H_{i} of 𝐻. In this paper, we complete the classification of the finite groups 𝐺 in which all subgroups of order p k p^{k} are ℳ-supplemented. In particular, we show that if k ≥ 2 k\geq 2 , then G / O p ′ ⁢ ( G ) G/\mathbf{O}_{p^{\prime}}(G) is supersolvable with a normal Sylow 𝑝-subgroup and a cyclic 𝑝-complement.

Brauer character degrees and Sylow normalizers

If p and q are primes, and G is a p-solvable finite group, it is possible to detect that a q-Sylow normalizer is contained in a p-Sylow normalizer using the character table of G. This is characterized in terms of the degrees of p-Brauer characters. Some consequences, which include yet another generalization of the Itô–Michler theorem, are also obtained.

Sylow branching coefficients and a conjecture of Malle and Navarro

We prove that a finite group G $G$ has a normal Sylow p $p$ -subgroup P $P$ if, and only if, every irreducible character of G $G$ appearing in the permutation character ( 1 P ) G $({\bf 1}_P)^G$ with multiplicity coprime to p $p$ has degree coprime to p $p$ . This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.

N-critical graph of finite groups

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

AbstractLet G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

Normal Sylow subgroups and monomial Brauer characters

Normal Sylow subgroups and monomial Brauer characters

English

Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent or subnormal. Moreover, G has either a normal Sylow p-subgroup or a normal p-complement for each prime divisor p of |G|.

Two results on the character degree sums

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T(G). We show that if |G|/T(G)<(p+1)/2, then G has a normal Sylow p-subgroup; and that if |G|/T(G)<3p2/(p2+2), then G is p-supersolvable, where p is a prime.