Related papers: Products of Differences in Prime Order Finite Fiel…

We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp/log_2 p$. We conjecture that we…

We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized…

It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant…

It is shown that finite groups in which the order of the product of every pair of elements of co-prime order is the product of the orders, is nilpotent.

Let $\mathbb{F}_p$ be a prime field of order $p,$ and $A$ be a set in $\mathbb{F}_p$ with $|A| \leq p^{1/2}.$ In this note, we show that \[\max\{|A+A|, |f(A, A)|\}\gtrsim |A|^{\frac{6}{5}+\frac{4}{305}},\] where $f(x, y)$ is a…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant…

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

Let $G$ be a group and $G_0 \subseteq G$ be a subset. A sequence over $G_0$ means a finite sequence of terms from $G_0$, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a…

Let A be a subset of positive relative upper density of P^d, the d-tuples of primes. We prove that A contains an affine copy of any finite set of lattice points E, as long as E is in general position in the sense that it has at most one…

For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $\mathbb{F}_{q}$ when $q> 2401$.

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains…

Every finite non-abelian group of order $n$ has a non-central element whose centralizer has order exceeding $n^{1/3}$. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…

Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least…

In the stable general linear group over an arbitrary field, we prove that every element with determinant $\pm 1$ is the product of three involutions, and of no less in general. We also obtain several results of the same flavor, with…

We obtain an exact formula for the average order of elements of a wreath product of two finite groups. Then focussing our attention on $p$-groups for primes $p$, we give an estimate for the average order of a wreath product $A\wr B$ in…

For a finite field of odd number of elements we construct families of permutation binomials and permutation trinomials with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Binomials and…