Related papers: On quantitative structure of symplectic groups

We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We…

A $P_4$-free graph is called a cograph. In this paper we partially characterize finite groups whose power graph is a cograph. As we will see, this problem is a generalization of the determination of groups in which every element has prime…

For $g\in \mathbb{N}$, let $G=\Sp(2g,\mathbb{Z})$ be the integral symplectic group and $S(g)$ be the set of all positive integers which can occur as the order of an element in $G$. In this paper, we show that $S(g)$ is a bounded subset of…

Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the…

We construct a solvable group G of order 5648590729620 such that the set of element orders of G coincides with that of the simple group S(4,3). This completes the determination of finite simple groups isospectral to solvable groups.

We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that…

In this paper, we completely determine the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle $W(q)$, $q$ odd. As a corollary, we show that the finite groups that act regularly on the points of a…

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

We consider the structure of a finite groups having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived…

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under the assumptions that (i) the order of the group is coprime to $p$ and (ii) either the…

We consider the four structures $(\mathbb{Z}; \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Z}; <, \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Q}; \mathrm{Sqf}^\mathbb{Q})$, and $(\mathbb{Q}; <, \mathrm{Sqf}^\mathbb{Q})$ where $\mathbb{Z}$ is the additive…

In this note, we give an alternative proof of the following result. Let p, q >= 2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

The reality of the difficulties in investigation of finite groups are considered. It is shown that the consideration of symmetry properties of the $k$-orbits that are obtained with an action of a finite group $F=(V,\cdot)$ on Cartesian…

We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that…

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer…

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected…

A strong Gelfand pair $(G,H)$ is a finite group $G$ together with a subgroup $H$ such that every irreducible character of $H$ induces to a multiplicity-free character of $G$. We classify the strong Gelfand pairs of the finite symplectic…

Let $\omega$ be a Kaehler form on the real $4$-torus $T^4$. Suppose that $\omega$ satisfies an irrationality condition which can be achieved by an arbitrarily small perturbation of $\omega$. This note shows that the smoothly trivial…