Xiuyun Guo

On Semi-Cover-Avoiding Maximal Subgroups and Solvability of Finite Groups

A subgroup H of a finite group G is said to be “semi-cover-avoiding in G” if there is a chief series of G such that H covers or avoids every chief factor of the chief series. In this article, some new characterizations for finite solvable groups are obtained based on the assumption that some subgroups have semi-cover-avoiding properties in the groups.

On ℋC-Subgroups and the Structure of Finite Groups

A subgroup H of a group G is said to be an ℋC-subgroup of G if there exists a normal subgroup T of G such that G = HT and H g ∩ N T (H) ≤ H for all g ∈ G. In this article, some new characterizations for p-nilpotency and supersolvability of finite groups are presented. In addition, we discuss the structure of ℋC*-group (a group G is called an ℋC*-group if every subgroup of G is an ℋC-subgroup of G) and prove that a group G is an ℋC*-group if and only if there is a nilpotent normal subgroup H of G such that G/H is abelian and each element of G induces a power automorphism on H by conjugation.

THE INFLUENCE OF MINIMAL SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

A subgroup H of a finite group G is called ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-quasinormal in K. In this paper, we investigate the influence of ss-supplemented minimal subgroups on the structure of finite groups and obtain some interesting results.

On s-Permutable Subgroups and p-Nilpotency of Finite Groups

A subgroup H of a finite group G is said to be s-permutable in G if H permutes with every Sylow subgroup of G. In this article, some sufficient conditions for a finite group G to be p-nilpotent are given whenever all subgroups with order p m of a Sylow p-subgroup of G are s-permutable for a given positive integer m.

On p-Regular G-Conjugacy Classes and the p-Structure of Normal Subgroups

Let N be a p-solvable normal subgroup of a group G such that N contains a noncentral Sylow r (≠ p)-subgroup R of G. It is proved that the p-complements of N are nilpotent if |x G |=1 or m for every p-regular element x of N whose order is divisible by at most two distinct primes. Our result, therefore, gives some information concerning the nilpotence of some kind of subgroups of a group G.

Finite Groups All of Whose Second Maximal Subgroups Are PSC ∗ -Groups

A subgroup H of a finite group G is called a QTI-subgroup if C G (x) ≤ N G (H) for any 1 ≠ x ∈ H. In this article, the finite groups all of whose second maximal subgroup are QTI-subgroups are classified.

On Thompson's Conjecture of A 10

In this short article, we prove the Thompsons Conjecture is true for the alternating group A 10, where A 10 has exactly one connected prime graph component.

Actions on p-groups with the kernel containing the 𝔉-residuals

ABSTRACT Let A be a finite group of odd order and let A act on a finite p-group P with |P|>pe, where e is an integer e≥4(e≥5 if p = 2). In this paper we show that P is centralized by if every non-meta-cyclic subgroup of order pe in P is stabilized by Op(A). As applications, some conditions are given for a finite group G with the p-length ≤1 and the p-rank ≤2. We also find a class of finite p-groups, which is not only very useful for the paper but also has its independent meaning.

Finite Groups with Small Normal Closure of Cyclic Subgroups

A group G is called an I(r)-group if | ⟨ x ⟩ G : ⟨ x ⟩ | divides the product of r primes for every x ∈ G. In this article, we mainly study the properties of finite I(1)-groups and I(2)-groups. In particular, we get the possible upper bound of the order of their derived subgroup of such groups.

Finite Groups with a Pre-Fixed-Point-Free Power Automorphism

A power automorphism θ of a group G is said to be pre-fixed-point-free if CG(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms.