With the help of some $p$-adic formal series over $p$-adic number fields and the estimates of character sums over Galois rings, we prove that there is a constant $C(n)$ such that there exists a primitive polynomial $f(x)\,{=}\,x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ of degree $n$ over $F_{q}$ with the first $m=\lfloor\frac{n-1}{2}\rfloor$ coefficients $a_{1},\ldots ,a_{m}$ prescribed in advance if $q\,{>}\,C(n)$.

We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products .

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

We compute the precise bounds, for every positive integer n, for the nilpotency class of nilpotent subgroups of {\text GL}(n,{\tf="P7DC6"Q}) and {\text GL}(n,{\tf="P7DC6"Z}).

It is conjectured that the central quotient of any irreducible Artin–Tits group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin–Tits groups that are known to be CAT(0) groups by a result of Brady and McCammond, that is, Artin–Tits groups associated with graphs having no 3-cycles and Artin–Tits groups of almost large type associated with graphs admitting appropriate directions. In particular, the latter family contains Artin–Tits groups of large type associated with cones over square-free bipartite graphs.

Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p>0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.