Theta Pairs and the Structure of Finite Groups

We introduce the notion of -pair of a proper subgroup of a finite group and give necessary and sufficient conditions for supersolvability and nilpotence of a finite group. As application, we give a proof to a recent open problem of Skiba in the unsolved problems in group theory.     

A class of periodic groups

We deal with periodic groups saturated with dihedral groups. In particular, it is proved that periodic groups of bounded period, and also periodic Shunkov groups, saturated with dihedral groups, are locally finite.Supported by RFBR grant No. 03-01-00356 and by the Krasnoyarsk Science Foundation, project 11F0202C.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 114–125, January–February, 2005.     

Quantum group invariant fermionic gases: GLp,q(2) and SUp/q(2) invariances

We first investigate the high-temperature behavior of a two-parameter deformed quantum group fermionic gas with GL_p,q(2) symmetry, where (p,q)C×C. We then discuss both the structural and thermodynamical differences between GL_p,q(2)- and SU_r(2)-fermions with r=p/q where (p,q)R×R.     

On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

A rank-three condition for invariant (1, 2)-symplectic almost Hermitian structures on flag manifolds

This paper considers invariant (1, 2)-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group G. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of G, and generalizes the cone-free property for tournaments related to 𝕊l (n,ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not B _l , l ≥ 3, G ₂ or F ₄. For B _l and F ₄ a close condition turns out to be sufficient. Received: 28 October 2001