On the Frattini lemma

Let $K$ be a subgroup of a finite group $G$, and suppose that $G=KN_G(P)$ for every Sylow subgroup $P$ of $K$. Then the subgroup $K$ is normal in $G$

Finite Groups with Hall Schmidt Subgroups

A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. We study the properties of a non-nilpotent group G in which every Schmidt subgroup is a Hall subgroup of G.Comment: 9 page

On Maximal Subgroups of a Finite Solvable Group

The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in \pi(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$

Superalgebraic structure of Lorentz transformations

Modern relativistic theory of the second quantization of fermion and boson fields is based on the use of the mathematical apparatus of C*-algebras and Lie superalgebras. In this case, for fermions, the Lorentz transformations are considered as Bogolyubov transformations of creation and annihilation operators. However, in this approach one can not obtain an explicit form of the Dirac gamma-matrices. The mathematical apparatus of the superalgebraic representation of the algebra of the second quantization of spinors is developed in the article. It is based on the use of density in the impulse space of Grassmann variables and their derivatives. It is shown that the Dirac matrices and the Lorentz transformation generators can be expressed in terms of such densities. A superalgebraic form of the Dirac equation and the vacuum state vector are constructed. It is shown that in the superalgebraic form of the complex Clifford algebra the generators corresponding to the Dirac gamma matrices are not equivalent. Clifford vector corresponding to diagonal matrix annihilates the vacuum, and the remaining ones give nonzero values. This means that there is asymmetric direction corresponding to the time axisComment: Preprint of Proc. of Intern. Scientific Meeting PIRT-2017. Moscow, 3-6 July, 201

On a finite group having a normal series whose factors have bicyclic Sylow subgroups

We consider the structure of a finite groups having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.Comment: 9 page

A note on the supersolvability of a finite group with prime index of some subgroups

In this paper, we proved that a group $G$ is supersoluble if and only if for any prime $p\in \pi (G)$ there exists a supersoluble subgroup of index $p$

Finite soluble groups with nilpotent wide subgroups

A subgroup of a finite group is wide if each prime divisor of the group order divides the subgroup order. We obtain the description of finite soluble groups with no wide subgroups. We also prove that a finite soluble group with nilpotent wide subgroups has the quotient group by its hypercenter with no wide subgroups

Construction of the fermionic vacuum and of fermionic operators of creation and annihilation in the theory of algebraic spinors

We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.Comment: 4 pages, Russian languag

Electron Screening in the 7Be + p -->8B + photon reaction

We evaluate the effect of screening by bound electron in the 7Be + p --> 8B + photon reaction, where 7Be target contains bound electron, in the framework of the adiabatic representation of the three particle problem. A comparison with two other approximations (united atom and folding) is presented. A good agreement between the ``united atom'' approximation and the exact solution is found. We also discuss the screening corrections induced by two K-shell electrons on a 7Be target. The bound electron screening effect consequences for 7Be and 8B solar neutrino fluxes are discussed.Comment: Revised paper. (RevTeX, 6 pages, 2 PS-figures

Finite groups with P\mathbb P-subnormal primary cyclic subgroups

A subgroup $H$ of a group $G$ is called $\mathbb P$-subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset ... \subset H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for all $i$. In this paper, we study the groups in which all primary cyclic subgroups are $\mathbb P$-subnormal