# Wang Youyu

### Groups all cyclic subgroups of which are BN A-subgroups

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 284-288

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is a BN A-subgroup of $G$ if either $H^x = H$ or $x \in \langle H, H^x\rangle$ for all $x \in G$. The BN A-subgroups of $G$ are between normal and abnormal subgroups of $G$. We obtain some new characterizations for finite groups based on the assumption that all cyclic subgroups are BN A-subgroups.

### Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions

Bai Yongzhen, Li Yannan, Wang Youyu

Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1646–1656

We establish some new Lyapunov-type inequalities for one-dimensional *p*-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented.

*c *^{*} -Supplemented subgroups and *p *-nilpotency of finite groups

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1011–1019

A subgroup $H$ of a finite group $G$ is said to be $c^{*}$-supplemented in $G$ if there exists a subgroup $K$ such that $G = HK$ and $H ⋂ K$ is permutable in $G$. It is proved that a finite group $G$ that is $S_4$-free is $p$-nilpotent if $N_G (P)$ is $p$-nilpotent and, for all $x ∈ G \backslash N_G (P)$, every minimal subgroup of $P ∩ P^x ∩ G^{N_p}$ is $c^{*}$-supplemented in $P$ and (if $p = 2$) one of the following conditions is satisfied:

(a) every cyclic subgroup of $P ∩ P^x ∩ G^{N_p}$ of order 4 is $c^{*}$-supplemented in $P$,

(b) $[Ω2(P ∩ P^x ∩ G^{N_p}),P] ⩽ Z(P ∩ G^{N_p})$,

(c) $P$ is quaternion-free, where $P$ a Sylow $p$-subgroup of $G$ and $G^{N_p}$ is the $p$-nilpotent residual of $G$.

This extends and improves some known results.