Xiuyun Guo

Let [ ] be a nontrivial finite group and denote by [ ] the set of all pairs [ ] with nontrivial subgroups [ ] and the irreducible complex character [ ] of [ ]. We say that [ ] in [ ] if [ ] and [ ] is a constituent of [ ]. The connectivity of the poset [ ] is studied in this paper. In fact, we prove that [ ] is disconnected if and only if [ ] is a cyclic group of prime power order or a quaternion group.

A proper subgroup [ ] of a finite group [ ] is called a second maximal subgroup of [ ] if [ ] is a maximal subgroup of every maximal subgroup [ ] in [ ] with [ ]. In this paper, we investigate the structure of the finite group [ ] in which [ ] is a second maximal subgroup for every non-central element [ ] in [ ], and prove that either [ ] is solvable or [ ], where [ ] with [ ] a prime or [ ] with [ ] an odd prime.

A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

A subgroup H of a group G is said to be a BNA-subgroup of G if either H^{x}= H or x \in \langle H, H^{x}\rangle for all x\in G . The purpose of this paper is first to give the best bound for the Fitting height of G if all minimal subgroups of G are BNA-subgroups of G , and next to give an answer to the question of He, Li, and Wang [Rend. Semin. Mat. Univ. Padova 136 (2016), 51–60]. Finally, we use a few BNA-subgroups of prime order to determine the structure of the finite groups. In fact, some new conditions for a finite group to be supersolvable have been given.

Let [ ] be a finite [ ]-group. Then [ ] is said to be a [ ]-group if [ ] for every nonnormal subgroup [ ] of [ ]. In this paper, we study the [ ]-group [ ] and get [ ]. It is proved that [ ] if [ ] and [ ] if [ ].