suppose that  is a finite group. then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . suppose that . then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . in this paper, we prove that if  is a group such that , where , then . here  denotes the family of suzuk...

let $g$ be a finite group. we construct the prime graph of $ g $,which is denoted by $ gamma(g) $ as follows: the vertex set of thisgraph is the prime divisors of $ |g| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ g $ contains anelement of order $ pq $.in this paper, we determine finite groups $ g $ with $ gamma(g) =gamma(l_3(q)) $, $2 leq q < 100 $ and prov...

On the projective plane there is a unique cubic root of canonical bundle and this acyclic. fake planes such exists if are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any must be In present note give two short proofs statement show acyclicity some other line bundles on with at least $9$ automorphisms. Similarly to our earlier work employ sim...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. particular, these have infinite w-verbal width for all proper words w.

We consider WZW models based on the non-semi-simple algebras that they were recently constructed as contractions of corresponding algebras for semi-simple groups. We give the explicit expression for the action of these models, as well as for a generalization of them, and discuss their general properties. Furthermore we consider gauged WZW models based on these non-semi-simple algebras and we sh...

We study the base sizes of finite quasiprimitive permutation groups twisted wreath type, which are precisely with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every group type isomorphic to product G=Tk:P acting on its Ω=Tk, where T some non-abelian simple P transitively k={1,…,k} k⩾2. prove if G primitive Ω k, then has size 2. proportion pairs points bases...

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an compact Hausdorff space are continuously orbit equivalent only if they conjugate. also show above fails we replace by certain other virtually cyclic groups, e.g., direct product integer with non-abelian finite simple group.

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

Abelian cellular automata (CA) are CA which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images weak∗-converge towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i.e. randomization for a wide class of init...