Long Miao

Yangzhou University · college of mathematics

A subgroup H of a finite group G is called a p-nilpotent quotient-supplement in G if there exists a subgroup K of G such that G = HK and K/K ∩ H G is p-nilpotent. By using this concept, we establish some new criteria for a finite group G to be p-nilpotent.

A subgroup H of G is called M-supplemented in G if there exists a subgroup B of G such that G = HB and H i B < G for every maximal subgroup H i of H. In this paper, we use p-supplemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results i...

Let G be a finite group and \(\mathscr {F}\) a saturated formation of finite groups. Then G is a quasi-\(\mathscr {F}\)-group if for every \(\mathscr {F}\)-eccentric chief factor H / K of G and every \(x\in G\), x induces an inner automorphism on H / K. In this article, we obtain some results about the quasi-\(\mathscr {F}\)-groups and use them to...

A subgroup H of a finite group G is called a p-nilpotent quotient-supplement in G if there exists a subgroup K of G such that G = HK and K/K ∩ H G is p-nilpotent. By using this concept, we establish some new criteria for a finite group G to be p-nilpotent.

A subgroup H of G is called complemented in G if there exists a subgroup K of G such that and . The aim of this paper is to prove the following: A finite group G is solvable if and only if its Sylow 3-, 5- and 7-subgroups are complemented in G.

Suppose that A is a subgroup of a group G. A is called to be m-embedded in G if G has a subnormal subgroup T and a \(\{1\le G\}\)-embedded subgroup C such that \(G=AT\) and \(A\cap T\le C\le A\). In this paper, we shall investigate the structure of finite groups by using m-embedded subgroups and obtain some new characterization about p-supersolvabi...

A subgroup A of a finite group G is called {1 ≤ G}-embedded in G if for each two subgroups K ≤ H of G, where K is a maximal subgroup of H, A either covers the pair (K, H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1 ≤ G}-embedded subgroup C such that G = HT and H ∩ T ≤ C ≤ H. In this pape...

A subgroup H of a group G is said to be \(\mathcal {M}^{*}\)-supplemented in G if there exists a subgroup K of G such that G = H K and H∩K is \(\mathcal {M}\)-supplemented in G. In this paper, we prove as follows: Let E be a normal subgroup of a group G. Suppose that every maximal subgroup of every non-cyclic Sylow subgroup P of F ∗(E) is \(\mathca...

Let G be a finite group, p a prime and p a Sylow p-subgroup of G. The main goal of the paper is to determine sufficient conditions for the p-nilpotence of a finite group with p-nilpotent Sylow normalizers.

Let A be a subgroup of G. A is m-embedded in G if G has a subnormal subgroup T and a {1 < G}-embedded subgroup C such that G = AT and T ∩ A < C < A. In this paper, we study the structure of finite groups by using m-embedded subgroups and obtain some new results about p-supersolvability and p-nilpotency of finite groups.

Let G be a finite group and let ℱ be a class of groups. Then Z ℱφ(G) is the ℱφ-hypercentre of G, which is the product of all normal subgroups of G whose non-Frattini G-chief factors are ℱ-central in G. A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and H1B is a proper subgroup of G for a...

A subgroup H is called nearly ℳ-supplemented in a finite group G if there exists a normal subgroup K of G such that HK ⊴ G and TK < HK for every maximal subgroup T of H. We obtain some new results on supersoluble groups and their formation by using nearly ℳ-supplemented subgroups and study the structure of finite groups.

A subgroup K of G is M p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p α . In this paper we prove the following: Let p be a prime divisor of |G| and let H be a p-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D p ≠ 1 and |H:...

Let d be the smallest generator number of a finite p-group P and M-d(P) = {P-1,P-2, . . . P-d} be the set of maximal subgroups of P such that boolean AND(d)(i=1) P-iota = Phi(P). Then P is called M-permutable in a finite group G, if there exists a subgroup B of G such that G = PB and PiB < G for every P-i of M-d(P). In this paper, we investigate th...

A subgroup H of a group G is called Q-permutable in G if there exists a subgroup B of G such that (1) G = HB and (2) if H 1 is a maximal subgroup of H containing H QG , then H 1B = BH 1 < G, where H QG is the largest permutable subgroup of G contained in H. In this paper, we prove the following statement: Let \( \mathcal{F} \) be a saturated format...

Let G be a finite group and F a formation of finite groups. A subgroup H of G is called F s - supplemented in G if there exists a subnormal subgroup T of G such that G = HT and (H ∩T)H G/H G is contained in the F -hypercenter Z Finfin; (G/H G) of G/H G . In this paper, we study the structureof finite groups by using F s-supplemented subgroups.

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)HG/HG is contained in the ℱ-hypercenter of G/HG. In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated...

A subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of...

A subgroup H is called s-supplemented in G, if there exists a subgroup K of G such that G=HK and H∩K≤H SG , where H SG is the largest subnormal subgroup of G contained in H. In this paper, we investigate the influence of s-supplemented primary subgroups in finite groups. Some new results about p-nilpotency of finite groups are obtained.

Let d be the smallest generator number of a finite p-group P, and let M(d)(P) = {P(1), P(2),..., P(d)) be a set of maximal subgroups of P such that boolean AND(d)(i=1) P(i) = Phi(P). Then P is called M-permutable in a finite group G, if there exists a subgroup B of G such that G = PB and P(i)B < G for any P(i) of M(d)(P). In this article, we invest...

Let 𝔉 be a class of finite groups. A subgroup H of G is said to be 𝔉-supplemented in G if there exists a subgroup T of G such that G = TH and (H ∩ T)HG/HG is contained in the 𝔉-hypercenter of G/HG. In this paper, we investigate further the influence of 𝔉-supplemented subgroup on the structure of finite groups and generalize a series of known result...

A subgroup H of a group G is said to be weakly s-permutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H sG where H sG is the largest s-quasinormal subgroup of G contained in H. In this paper, we investigate the influence of weak s-permutability of some primary subgroups in finite groups. Some new results about...

A subgroup H of a group G is said to be weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H 1/H G is a maximal subgroup of H/H G, then H 1B = BH 1 <G. where H G is the largest normal subgroup of G contained in H. In this paper we will prove that: Let F be a saturated formation containing all supersolv...

A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H G where H G is the largest normal subgroup of G contained in H. In this pape...

A subgroup H is called ℳ-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H 1 B is a proper subgroup of G for every maximal subgroup H 1 of H. We investigate the influence of ℳ-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.

A subgroup H of a finite group G is called to have semi cover-avoiding property in G if there is a normal series of G such that H either covers or avoids every normal factor of the series. In this article we get some new results under the assumption that every maximal subgroup of Sylow subgroups of a suited subgroup of G has semi cover-avoiding pro...

A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and such that H1B is a proper subgroup of G for any maximal subgroup H1 of H. In this paper we fix a subgroup D in every non-cyclic Sylow subgroup P of G satisfying 1 < D < P and study the structure of G under the assumption that all subgrou...

A subgroup H is called ℳ-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H1B is a proper subgroup of G for any maximal subgroup H1 of H. In this article, we investigate the influence of ℳ-supplementation of some primary subgroups in finite groups. Some new results about supersolvable groups and formation are...

A p-subgroup P ≠ 1 of G is called -permutable in G if there exists a set of maximal subgroup Pi of P and a subgroup B of G such that: (1) ⋂i=1d Pi = Φ(P) and |P : Φ(P)| = pd; (2) G = P B and Pi B = B Pi < G for any Pi of . In this paper, we investigate the influence of -permutability of Sylow subgroups in finite groups. Some new results about super...

Let \( {\user1{\mathcal{F}}} \) be a class of groups. A subgroup H of a group G is called \( {\user1{\mathcal{F}}} \)-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ H G belongs to \( {\user1{\mathcal{F}}} \) where H G is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to...

We call a subgroup H of a group G p-nilpotent quotient-supplemented in G if there exists a subgroup K of G such that G = HK and K/K ∩ H G is p-nilpotent. By using this concept, we establish some new criteria for a group G to be p-nilpotent.

A subgroup H of a group G is said to be ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ℱ be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ℱ. Suppose that every maxima...

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.

Let F be a class of groups. A subgroup H of a group G is called F-s-supplemented in G if there exists a subgroup K of G such that G = HK and K|K ∩ HG belongs to F. We obtain some results about the F-s-supplemented subgroups and use them to determine the structure of some groups. In particular, some new criteria of p-nilpotency, solubility, supersol...

We use complements of primary subgroups to give some sufficient conditions under which a group belongs to a given local formation. Some results on complemented subgroups are obtained.