assume that $(n,l)$, is a pair of finite dimensional nilpotent lie algebras, in which $l$ is non-abelian and $n$ is an ideal in $l$ and also $mathcal{m}(n,l)$ is the schur multiplier of the pair $(n,l)$. motivated by characterization of the pairs $(n,l)$ of finite dimensional nilpotent lie algebras by their schur multipliers (arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

A finite abelian group G of order n is said to be of type III if all divisors of n are congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of cardinality very “close” to the largest possible in a finite abelian group G of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality i...

We introduce and discuss the properties of a new class of finite-dimensional frames with strong symmetry properties called geometrically uniform (GU) frames, that are defined over a finite abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by a finite abelian group of unitary...

We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of...

An infinite group with supersimple theory has a finite series of definable groups whose factors are infinite and either virtually-FC or virtuallysimple modulo a finite FC-centre. We deduce that a group which is typedefinable in a supersimple theory has a finite series of relatively definable groups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple ...

There is a longstanding conjecture of Nussbaum, which asserts that every finite set in R on which a cyclic group of sup-norm isometries acts transitively contains at most 2 points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold more generally for abelian groups of sup-norm is...

Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field A: to a category of Z-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of Z-modules. We use Deligne's equivalence to characterize the finite group schemes over k that occur as kernels of polarizatio...

the non commuting graph of a non-abelian finite group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we prove some new results about this graph. in particular we will give a new proof of theorem 3.24 of [2]. we also prove that if $g_1$, $g_2$, ..., $...

A connection is developed between polynomials invariant under abelian permutation of their variables and minimal zero sequences in a finite abelian group. This connection is exploited to count the number of minimal invariant polynomials for various abelian groups.

the triple factorization of a group $g$ has been studied recently showing that $g=aba$ for some proper subgroups $a$ and $b$ of $g$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. in this paper we study two infinite classes of non-abelian finite groups $d_{2n}$ and $psl(2,2^{n})$...