On the <Emphasis Type="Italic">p</Emphasis>-supersolubility of a finite group with a µ-supplemented sylow <Emphasis Type="Italic">p</Emphasis>-subgroup

A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H ₁ K is a proper subgroup in G for every maximal subgroup H ₁ in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.     

Finite groups in which Sylow normalizers have nilpotent Hall supplements

The normalizer of each Sylow subgroup of a finite group G has a nilpotent Hall supplement in G if and only if G is soluble and every tri-primary Hall subgroup H (if exists) of G satisfies either of the following two statements: (i) H has a nilpotent bi-primary Hall subgroup; (ii) Let π(H) = {p, q, r}. Then there exist Sylow p-, q-, r-subgroups H _p , H _q , and H _r of H such that H _q ? N _H (H _p ), H _r ? N _H (H _q ), and H _p ? N _H (H _r ).     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

X-quasinormal subgroups

Considering two subgroups A and B of a group G and ? ≠ X ? G, we say that A is X-permutable with B if AB ^x = B ^x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

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and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups with a class of complemented normal subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

Classification of finite groups satisfying a minimal condition

If H is a subgroup of a finite group G then we denote the normal closure of H in G by H ^G . We call G a PE-group if every minimal subgroup X of G satisfies N ^G (X) ∩ X ^G = X. The authors classify the finite non-PE-groups whose maximal subgroups of even order are PE-groups.     

Every 2-group with all subgroups normal-by-finite is locally finite

A group G has all of its subgroups normal-by-finite if H/H _G is finite for all subgroups H of G. The Tarski-groups provide examples of p-groups (p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |H/H _G | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.     

<Emphasis Type="Italic">c</Emphasis>-Semipermutable subgroups of finite groups

A subgroup is called c-semipermutable in G if A has a minimal supplement T in G such that for every subgroup T ₁ of T there is an element x ∈ T satisfying AT ₁ ^x = T ₁ ^x A. We obtain a few results about the c-semipermutable subgroups and use them to determine the structures of some finite groups.     

Finite groups with ℱ-subnormal conditions

Let ? be a subgroup-closed saturated formation. A finite group G is called an ?_pc-group provided that each subgroup X of G is ?-subabnormal in the ?-subnormal closure of X in G. Let ?_pc be the class of all ?_pc-groups. We study some properties of ? _pc-groups and describe the structure of ?_pc-groups when ? is the class of all soluble π-closed groups, where π is a given nonempty set of prime numbers.     

A criterion for the <Emphasis Type="Italic">F</Emphasis> <Subscript> <Emphasis Type="Italic">π</Emphasis> </Subscript>-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.     

Finite p-Groups with Few Non-major k-Maximal Subgroups

A subgroup of index p ^k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ _k (G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δ_d(G)(G) ≤ p² and δ_d(G)?1(G) = 0, respectively.     

On weakly <Emphasis Type="Italic">S</Emphasis>Φ-supplemented subgroups of finite groups

Let G be a finite group. We say that a subgroup H of G is weakly SΦ-supplemented in G if G has a subgroup T such that G = HT and H∩T ≤ Φ(H)H_sG, where H_sG is the subgroup of H generated by all those subgroups of H that are s-permutable in G. In this paper, we investigate the influence of weakly SΦ-supplemented subgroups on the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.     

Finite Groups with Given Weakly <Emphasis Type="Italic">σ</Emphasis>-Permutable Subgroups

Let G be a finite group and let σ = {σ _i | i ∈ I} be a partition of the set of all primes P. A set ? of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ? is a Hall σ _i -subgroup of G and ? has exactly one Hall σ _i -subgroup of G for every σ _i ∈ σ(G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ? such that HA ^x = A ^x H for all A ∈ ? and all x ∈ G. A subgroup H of G is said to be weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _σG , where H _σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. We study the structure of G under the condition that some given subgroups of G are weakly σ-permutable in G. In particular, we give the conditions under which a normal subgroup of G is hypercyclically embedded. Some available results are generalized.     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. 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; H is said to be weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup \(H_{se}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{se}\). 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 ss-quasinormal or weakly s-permutably embedded in G. Some recent results are generalized and unified.     

Clifford Theory for Glider Representations

Classical Clifford theory studies the decomposition of simple G-modules into simple H-modules for some normal subgroup H ? G. In this paper we deal with chains of normal subgroups 1?G ₁?· · ·?G _d = G, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field K and relate different representations of the groups appearing in the chain. Picking some normal subgroup H ? G one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.     

Finite 2-groups whose length of chain of nonnormal subgroups is at most 2

Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept: we say that H₁ < H₂ < ? < H_k is a chain of nonnormal subgroups of G if each H_i ? G and if |H_i : H_i?1| = p for i = 2, 3,…, k. k is called the length of the chain. chn(G) denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn(G) ? 2 are completely classified up to isomorphism.