Let G be a finite group with a non-abelian minimal normal subgroup N which is a direct product of the simple group X. The maximal subgroups of G which complement N and their conjugacy classes are parametrised in terms of certain homomorphisms taking values in AutX and satisfying particular conditions.

The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. We will determine the kernel of the map induced on the second rational cohomology by the second Johnson homomorphism in terms of the representation theory of the symplectic group.

Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ the intersection graph of $G$, whose vertices are proper nontrivial subgroups with distinct joined by an edge if and only they intersect nontrivially. We prove that diameter has tight upper bound 5, thereby resolving question posed Shen (2010). Furthermore, 5 is achieved baby monster group certain unitary groups odd prime...

A simple algebraic model of charged particle coupled to singular magnetic field is given. Quantization is described as gradation by certain abelian group G. Statistics is determined by a commutation factor λ on the grading group G. Composite fermions and composite bosons are described in an unified way as Z 2 ⊕ ... ⊕ Z 2-graded λ-commutative algebras.

‎There are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{Z}_{2}$ or $mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this...

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disorderi...