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For a finite group G, a subgroup P of G is 2-minimal if B P , where B = N G ( S ) for some Sylow 2-subgroup S of G, and B is contained in a unique maximal subgroup of P. For fields of odd characteristic, this paper contains a detailed and explicit description of all the 2-minimal subgroups of the finite general orthogonal groups, and certain of their subgroups.

If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group., 12 pages

In a recent paper, Kishi and Komatsu have shown that the complex quadratic fields of discriminant $$\begin{aligned} \Delta _{18} = 4 - 3^{18n+3} \end{aligned}$$ have a class group whose 3-Sylow subgroup is of rank at least three for all integers n. The Kishi and Komatsu paper proves this by looking at the structure of the algebraic fields involved. Our purpose in this paper is to show that the class groups for the order whose discriminant is $$\Delta _{18}$$ have a 3-SSG of rank at least three, to show this using elementary means that would have been available to Gauss, and to elucidate further the group structure of the subgroup of classes of order 3. More generally, we shall show that $$\Delta _{18}$$ is actually a special case of a more general sequence of discriminants, all of which can be shown parametrically to have a class group whose 3-Sylow subgroup has rank at least three.

Throughout this paper, all groups are finite and G always denotes a finite group. Let σ = { σ i | i ∈ I } be a partition of the set of all primes P . The group G is said to be: σ-primary if G is a σ i -group for some i = i ( G ) ; σ-nilpotent if G = G 1 × … × G n for some σ-primary groups G 1 , … , G n ; σ-soluble if every chief factor of G is σ-primary; σ-full if G possesses a Hall σ i -subgroup for all i such that σ i ∩ π ( G ) ≠ ∅ . A subgroup A of G is said to be σ-permutable in G provided G is σ-full and A permutes with every Hall σ i -subgroup H of G, that is, A H = H A for all i; G is said to be a PσT-group if σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G. Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F . We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups. This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ-soluble PσT-groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group.

We study fields of values of the restriction to Sylow subgroups of irreducible characters of the normalizers of Sylow subgroups in symmetric and alternating groups. As an application, we show that these classes of groups admit a McKay bijection that respects fields of values and restriction to Sylow subgroups.

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8., Comment: 19 pages

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 ) 3 p 2 / ( p 2 + 2 ) , then G is p-supersolvable, where p is a prime.

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 lPlOp(G)l ≤ p or G has a composition factor isomorphic to PSL2(q), where q = pf , f ≥ 2 and Op(G) denotes the largest normal p-subgroup of G.

We prove that if the average number of Sylow subgroups of a finite group is less than $${{41} \over 5}$$ and not equal to $${{29} \over 4}$$ , then G is solvable or G/F(G) ≅ A5. In particular, if the average number of Sylow subgroups of a finite group is $${{29} \over 4}$$ , then G/N ≅ A5, where N is the largest normal solvable subgroup of G. This generalizes an earlier result by Moreto et al.

Let G be a finite group, and let P be a Sylow p-subgroup of G. Under the hypothesis that NG(P) is p-nilpotent, we provide some conditions to give a p-nilpotency criterion of finite groups by Engel condition, which improves some recent results.

Let G be a finite group and denote by $${\mathbb {Z}}G$$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided $${\mathbb {Z}}G \cong {\mathbb {Z}}H$$ and G is q-constrained. If additionally $$O_{q'}(G)$$ is soluble the set $$\mathrm{sn}(G)$$ of all Sylow numbers of G is determined by $${\mathbb {Z}}G.$$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table $$\mathrm{X}(G)$$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components $${\mathbb {Z}}G$$ determines $$\mathrm{sn}(G).$$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated.

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G: ker φ|p′ = φ(1)p′ for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p′ is the p′-part of φ(1).

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G / O p ′ ( G ) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.

The structure of finite groups of 2-rank 1 is largely defined by the classical Burnside and Brauer–Suzuki theorems. Burnside proved that all elements of odd order of a finite group with a cyclic 2-Sylow subgroup form a normal subgroup. S.I. Adyan showed that this statement does not hold in the class of periodic groups even in the case where a Sylow 2-subgroup has order 2 and coincides with the center of the group. The results of Burnside, Brauer, and Suzuki can be formulated as one theorem: in a finite group $$G$$ of 2-rank 1, the image of any involution in the quotient group $$G/O(G)$$ lies in the center of this quotient group. It is unknown whether the same statement holds for a periodic group $$G$$ (V.P. Shunkov’s Question 4.75 from The Kourovka Notebook). There is no answer even when the centralizer of an involution $$i$$ is a locally cyclic group (V.D. Mazurov’s Question 15.54 from The Kourovka Notebook). In Theorem 1, we give a partial affirmative answer to Question 4.75 under an additional condition: in the group $$G$$ an involution $$i$$ generates a finite subgroup with any element of order not divisible by 4. In particular, Question 4.75 is solved positively in the classes of binary finite and conjugate binary finite groups. In Theorem 2, we study the structure of a nonlocally finite group $$G$$ with a finite involution and an involution $$i$$ whose centralizer is a locally cyclic 2-group. An involution $$i$$ of a group $$G$$ is called finite if the subgroup $$\langle i,i^{g}\rangle$$ is finite for every $$g\in G$$ . In particular, Theorem 2 defines the structure of a counterexample (under the assumption of its existence) to Question 15.54. .

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|.

A subgroup A is seminormal in a finite group G if there exists a subgroup B such that G = AB and AX is a subgroup for each subgroup X from B. We study a group G = G1G2 . . .Gn with pairwise permutable supersolvable groups G1, . . . ,Gn such that Gi and Gj are seminormal in GiGj for any i, j ∈ {1, . . . , n}, i ≠ j. It is stated that GU = (G')N. Here N and U are the formations of all nilpotent and supersolvable groups, and HX and H' are the X-residual and the derived subgroup, respectively, of a group H. It is proved that a group G = G1G2 . . .Gn with pairwise permutable subgroups G1, . . .,Gn is supersolvable provided that all Sylow subgroups of Gi and Gj are seminormal in GiGj for any i, j ∈ {1, . . . , n}, i ≠ j.

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. 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 s∗-semipermutable in G. Some recent results are generalized and unified.

A subgroup S of 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 give some new characterizations of p-solvab...

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G. If G is non-nilpotent, then the structure of G has been determined. If G is nilpotent, then the structure of G is determined by the structure of its Sylow subgroups. However, the classification of finite metahamiltonian p-groups is an unsolved problem. In this paper, finite metahamiltonian p-groups are completely classified up to isomorphism.

Let p > 3 and q ≠ p be primes, let G be a finite q-solvable group and let P ∈ Syl p ( G ) . Then G has a unique irreducible q-Brauer character of p ′ -degree lying over 1 P if and only if N G ( P ) / P is a q-group. One direction of this result follows from a natural McKay bijection of p ′ -degree irreducible q-Brauer characters, which is obtained under suitable conditions.

Let G be a group. If for every proper normal subgroup N and element x of G with N〈x〉≠G, N〈x〉 is an FC-group, but G is not an FC-group, then we call G an NFC-group. In the present paper we consider the NFC-groups. We prove that every non-perfect NFC-group with non-trivial finite images is a minimal non-FC-group. Also we show that if G is a non-perfect NFC-group having no nontrivial proper subgroup of finite index, then G is a minimal non-FC-group under the condition “every Sylow p-subgroup is an FC-group for all primes p”. In the perfect case, we show that there exist locally nilpotent perfect NFC-p-groups which are not minimal non-FC-groups and also that McLain groups $M(\mathbb {Q},GF(p))$ for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p-groups such that the normalizer of every element of order p is an FC-subgroup.

Let G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

A finite group G is called a PST-group (respectively, PT-group, T-group) if every subnormal subgroup of G is Sylow permutable (respectively, permutable, normal) in G. Let F be a class of group and ...

We finish off the classification of the endo-trivial modules of finite groups with Sylow 2-subgroups isomorphic to a semi-dihedral 2-group started by Carlson, Mazza and Thevenaz in their article Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup published in 2013.

Let G be a finite group. Roughly speaking, a subgroup A of G is called a local partial covering subgroup if A G = A B for a suitable maximal G-invariant subgroup B of A G . Our main result is as follows: Let p d be a prime power with p ≤ p d ≤ | G | p , assume that all subgroups of p d , and all cyclic subgroups of order 4 when p d = 2 and a Sylow 2-subgroup of G is nonabelian, of G are local partial covering subgroups. Then G / O p ′ p ( G ) is p-supersolvable; furthermore if G is not p-supersolvable, then O p ′ p ( G ) / Φ is a homogeneous F p [ G ] -module where Φ / O p ′ ( G ) = Φ ( G / O p ′ ( G ) ) , and each irreducible constituent has dimension dividing d − log p ⁡ | Φ | p .

A subgroup A of a finite group G is called a local covering subgroup of G if AG = AB for all maximal G-invariant subgroup B of AG = 〈Ag, g ∊ G〉. Let p be a prime and d be a positive integer. Assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd = 2 and a Sylow 2-subgroup of G is nonabelian, of G are local covering subgroups. Then G is p-supersolvable whenever pd = p or $${p^d} \le \sqrt {{{\left| G \right|}_p}} $$ or pd ≤ ∣Op’p(G)∣p/p.

We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $${{29} \over 4}$$ then G is solvable or G/F(G) ≌ A5. This generalizes an earlier result by the third author.

Given an arbitrary partition $ \sigma $ of the set of primes, we give some criteria for the $ \sigma $ -subnormality of a Sylow subgroup of a finite group.

Let ℙ be the set of all prime numbers. A subgroup H of a finite group G is called ℙ-subnormal if either H = G or there exists a chain of subgroups H = H0 ≤ H1 ≤ … ≤ Hn = G such that |Hi : Hi − 1| ∈ ℙ, 1 ≤ i ≤ n. We prove that any finite group with ℙ-subnormal Sylow p-subgroup of odd order is p-solvable and any group with ℙ-subnormal generalized Schmidt subgroups is metanilpotent.

Let $ \pi_{x} $ be the set of primes greater than $ x $ . We prove that for all $ x\in{𝕉} $ the classes of finite groups $ D_{\pi_{x}} $ and $ E_{\pi_{x}} $ coincide; i.e., a finite group $ G $ possesses a $ \pi_{x} $ -Hall subgroup if and only if $ G $ satisfies the complete analog of the Sylow Theorems for a $ \pi_{x} $ -subgroup.

Let G be a finite group. A minimal subgroup of G is a subgroup of prime order. A subgroup of G is called S-quasinormal in G if it permutes with each Sylow subgroup of G. A group G is called an MS-g...

The question of whether weakly Sylow-permutable subgroups of locally finite groups are serial is studied. An affirmative answer is obtained for many types of group, particularly groups satisfying finiteness conditions or with some degree of commutativity.

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.

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., Comment: Example 5.12 has been corrected and expanded

A subgroup 𝐴 of a group 𝐺 is called 𝑂𝑆-propermutable in 𝐺 if there is a subgroup 𝐵 such that 𝐺 = 𝑁𝐺(𝐴)𝐵, 𝐴𝐵 is a subgroup of 𝐺 and the subgroup 𝐴 permutes with all Schmidt subgroups of 𝐵. In this situation, the subgroup 𝐵 is called 𝑂𝑆-prosupplement to 𝐴 in 𝐺. In this paper, we proved the 𝑝-solubility of a finite group 𝐺 such that a Sylow 𝑝-subgroup of 𝐺 is 𝑂𝑆-propermutable in 𝐺, where 𝑝 > 5.

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$, where $q$ is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including a new infinite series of such groups. Our methods are based on the understanding of blocks of cyclic defect and Young tableaux combinatorics.

Let G be a finite group and H a subgroup of G. We say that H is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a generalized smooth group (GS-group) if [G/L] is totally smooth for every subgroup L of G of prime order. In this paper, we investigate the structure of G under the assumption that each subgroup of prime order is S-permutable if the maximal subgroups of G are GS-groups.

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 $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ 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 $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$., 51 pages. In this version, we corrected some typos

Let $$\mathfrak{F}$$ be a formation. Properties of the class $$\mathrm{w}^{*}\mathfrak{F}$$ of all groups $$G$$ for which $$\pi(G)\subseteq\pi(\mathfrak{F})$$ and the normalizers of all Sylow subgroups are $$\mathfrak{F}$$ -subnormal in $$G$$ are studied. In particular, it is established that this class is a formation closed with respect to taking Hall subgroups. Hereditary saturated formations $$\mathfrak{F}$$ coinciding with $$\mathrm{w}^{*}\mathfrak{F}$$ are found.

UDC 512.542 Let $\Bbb P$ be the set of all primes. A subgroup $H$ of a finite group $G$ is called $\Bbb P$-subnormal, if either $H = G$ or there exists a chain of subgroups $H = H_0\le H_1\le \ldots \le H_n = G$ such that $|H_i\colon H_{i-1}|\in \Bbb P,$ $1\le i\le n.$We prove that any finite group with a $\Bbb P$-subnormal Sylow $p$-subgroup of odd order is $p$-solvability and any group with $\Bbb P$-subnormal generalized Schmidt subgroups is metanilpotent.