let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

Given a finite group G, for all sufficiently large d and for each q > 3 there are symmetric designs and affine designs having the same parameters as PG(d, q) and AG(d, q), respectively, and having full automorphism group isomorphic to G.

If π is a finite symplectic translation plane, it is shown that any affine homology group is cyclic and has order dividing the order of the kernel homology group. This criterion provides a means to ensure that a given spread is not symplectic. Furthermore, a variety of symplectically paired André spreads are constructed.

‎this article presents a systematic study for structure of finite wavelet frames‎ ‎over prime fields‎. ‎let $p$ be a positive prime integer and $mathbb{w}_p$‎ ‎be the finite wavelet group over the prime field $mathbb{z}_p$‎. ‎we study theoretical frame aspects of finite wavelet systems generated by‎ ‎subgroups of the finite wavelet group $mathbb{w}_p$.

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups Gn that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his con...

The affine group schemes of automorphisms the multilinear r -fold cross products on finite-dimensional vectors spaces over fields characteristic not two are determined. Gradings by abelian groups these structures, that correspond to morphisms from diagonalizable into automorphisms, completely classified, up isomorphism.

In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be sep...

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension. In this paper we are interested in the following question. Question. Classify all smooth affine surfaces X/C which admit a proper morphism f : X → X with degree f > 1. In [5] and [18], a classification of smooth projective surfaces with a self-map of degree > 1 has been given. Th...

We study affine Jacobi structures on an affine bundle π : A→M . We prove that there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A = ⋃ p∈M Aff(Ap,R) of affine functionals. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited ...