A converse of Baer’s theorem

Schur’s classical theorem states that for a group $G$ , if $G/Z(G)$ is finite, then $G'$ is finite. Baer extended this theorem for the factor group $G/Z_n(G)$ , in which $Z_n(G)$ is the $n$ -th term of the upper central series of $G$ . Hekster proved a converse of Baer’s theorem as follows: If $G$ is a finitely generated group such that $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ denotes the $(n+1)$ st term of the lower central series of $G$ . In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$ . Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.     

The converse of Schur’s theorem

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of G being nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).     

A generalization of a converse to Schur’s theorem

Niroomand (Arch. Math. 94 (2010) 401–404) proved a converse to a theorem of Schur in the following sense. He proved that if G is a group such that [G, G] is finite and G/Z(G) is finitely generated, then G/Z(G) is finite, of order bounded above by [G, G] ^k where k is the minimal number of generators required for G/Z(G). Here, we give a completely elementary short proof of a further generalization.     

An extension of Jung’s theorem

Theorem. Let a set X?Rⁿ have unit circumradius and let B be the unit ball containing X. Put C =conv \(\bar X\) D =diam C (=diam X), k =dim C,d _i = √(2i + 2)/i. Then: (i) D∈[d_n, 2]; (ii) k≧m where m∈{2,3,...,n} satisfies D∈[d_m, d_m?1) (d_i decreases by i); (iii) In case k=m (by (ii), this is always the case when m=n), C contains a k-simplex Δ such that: (α) its vertices are on δB; (β) the centre of B belongs toint Δ; (γ) the inequalitiesλ _k (D) ≦l ≦D with $$\lambda _k (D) = D\sqrt {\frac{{4k - 2D^2 (k - 1)}}{{2 - (k - 2)(D^2 - 2)}}, D \in (d_k ,d_{k - 1} )} $$ are unimprovable estimates for length l of any edge of Δ.     

Semialgebraic version of Whitney’s extension theorem

Archiv der Mathematik - In this note we prove a semialgebraic counterpart of Whitney’s extension theorem.     

A further extension of Rotfel’d theorem

We prove an inequality for concave functions and partitioned matrices whose numerical ranges lie in a sector. This complements a theorem by E.Y. Lee concerning the positive semi-definite case.     

An extension of Fujita’s non-extendability theorem for Grassmannians

In this paper we study smooth complex projective varieties X containing a Grassmannian of lines ${{\mathbb G}(1, r)}$ which appears as the zero locus of a section of a rank two nef vector bundle E. Among other things we prove that the bundle E cannot be ample.     

A generalization of Baer’s theorem

In this paper, we studied the supersolvability of the product of two subgroups and got a generalization of Baer’s theorem.     

A generalization of Obata’s theorem

In a complete Riemannian manifold (M, g) if the hessian of a real-valued function satisfies some suitable conditions, then it restricts the geometry of (M, g). In this paper we characterize all compact rank-one symmetric spaces as those Riemannian manifolds (M, g) admitting a real-valued functionu such that the hessian ofu has at most two eigenvalues ?u and $ - \frac{{u + 1}}{2}$ under some mild hypotheses on (M, g). This generalizes a well-known result of Obata which characterizes all round spheres.     

A generalization of Jentzsch’s theorem

We study the asymptotic behavior of the roots of polynomials given by a linear summation method for partial sums of the Fourier series.     

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.     

A generalization of Forelli’s theorem

The purpose of this paper is to present a generalization of Forelli’s theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka (Kompleks. Anal. i Prilozh, 232–240, 2006) of 2005.     

An extension of Günther’s volume comparison theorem

The aim of this note if to give an extension of a classical volume comparison theorem for Riemannian manifolds with sectional curvature bounded above (see Günther, P. Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7, 78–93 (1960)). For the case of a n-dimensional simply connected complete Riemannian manifold with nonpositive sectional curvature our theorem states that the function tarea(S _t (p))/t ^n–2 is convex for every pM where S _t (p) denotes the sphere of radius t with center p. In view of area(S ₀ (p))=0, it is easy to see that our theorem implies the classical result. A similar result holds true for simply connected manifolds with sectional curvature bounded above by a negative constant.Research partially supported by Fondecyt Grant # 1000713 and by UTFSM Grant # 120023