A note on the conjugacy classes of non-cyclic subgroups

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7 Meng, W., Li, S. R. (2014). Finite groups with few conjugacy classes of non-cyclic subgroups. Scientia Sin. Math. 44:939–944.[Crossref] , [Google Scholar]], Meng and Li showed the inequality δ(G)≥2^|π(G)|?2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2^|π(G)|?2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ?G?A₅ or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G.     

On division subrings normalized by almost subnormal subgroups in division rings

Periodica Mathematica Hungarica - Let D be a division ring with infinite center, K a proper division subring of D and N an almost subnormal subgroup of the multiplicative group $$D^*$$ of D. The...     

Full subgroups of unitary groups over division rings

The subgroups, full of transvections, of unitary groups over division rings with an involution are described. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 193–200, 1987.     

On free subgroups of units of rings

We prove that if for elements of a ring of characteristic zero and is not nilpotent, then there exists such that the group generated by and is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.

     

A note on subgroups in a division ring that are left algebraic over a division subring

Let D be a division ring with center F and K a division subring of D. In this paper, we show that a non-central normal subgroup N of the multiplicative group $$D^*$$ is left algebraic over K if and only if so is D provided F is uncountable and contained in K. Also, if K is a field and the n-th derived subgroup $$D^{(n)}$$ of $$D^{*}$$ is left algebraic of bounded degree d over K, then $$dim _FDle d^2$$ .     

A note on maximal subgroups of free idempotent generated semigroups over bands

We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.     

Are there free groups in division rings?

The following conjecture is investigated: a noncentral subnormal subgroup of the multiplicative group of a division ring contains a noncyclic free subgroup. Special cases are proved, entailing several known commutativity theorems. Also a new framework is presented for some kinds of commutativity theorems, based on the existence of (group) words for which one can always find an appropriate substitution by elements of such a subnormal subgroup that yields a noncentral element. Several families of such words are given; one gets commutativity theorems imposing some restrictions (like periodicity) to the image of these words.     

A note on the coefficient rings of polynomial rings

LetA andB be two reduced commutative rings with finitely many minimal prime ideals. If the polynomial algebrasA[X ₁ …X _n ]=B[Y ₁ …Y _n ] whereX _i ,Y _iF are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:A→B such thatB is finitely generated over ϕ(A).     

A note on non-congruence subgroups

Let Z (ω_d) be the ring of quadratic imaginary integers with discriminant d. Serre [SI] has demonstrated the existence of non-congruence subgroups of SL(2,Z (ω_d)). This note demonstrates how some of these subgroups can be found.     

A note on cosubnormal subgroups

Two subgroupsH andK of a groupG are called cosubnormal if they are both subnormal in the subgroup generated by them. In this paper some subnormality criteria for cosubnormal subgroups of nilpotent-by-abelian groups are given.     

A note on the existence of subgradients

We describe an apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space.Research partially funded on NSERC grant A5116.     

A note on quasi-Frobenius rings

It is shown that a semiperfect ring is quasi-Frobenius if and only if every closed submodule of is non-small, where denotes the direct sum of copies of the right -module and is the first infinite ordinal.

     

A note on perfect rings

The second author was partially supported by the National Science Foundation.     

A note on the modality of parabolic subgroups

For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod(R : X), is the maximal number of parameters on which a family of R-orbits on X depends upon.Let G be a simple algebraic group defined over an algebraically closed field K of characteristic 0. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical P_u via conjugation. The modality of P is defined as mod P mod(P : P_u).Let r and s be the semisimple rank of G and P respectively. We show that there is a quadratic polynomial ƒ with rational coefficients such that the modality of P is at least ƒ(r − s). In particular, the modality of a Borel subgroup B of G grows at least quadratically with r. As a consequence, we obtain a finiteness result for algebraic groups from [8]: there is only a finite number of simple algebraic groups admitting parabolic subgroups of prescribed semisimple rank and prescribed modality. Combining our lower bounds with upper bounds from [6], we can compute the modality of Borel subgroups in some small rank cases.     

A note on hereditary rings

Herediticity of tensor products of hereditary rings is described and used to characterize certain hereditary prime factor rings of polynomial rings over hereditary P.I. rings.     

A note on zip rings

A ring R is called right zip if the right annihilator r _R (S) of a subset S of R is zero, r _R (X)=0 for a finite subset X of S. In this note we will prove that for any u.p.-monoid M a right uniform ring R is right zip if and only if the monoid ring R[M] is right zip.