Related papers: Minimal Sets of Generators for Big Mapping Class G…

The plane Cremona group over the finite field $\mathbb{F}_2$ is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve…

K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter…

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in…

The finite topological quandles can be represented as $n\times n$ matrices, recently defined by S. Nelson and C. Wong. In this paper, we first study the finite topological quandles and we show how to use these matrices to distinguish all…

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A…

The phylogenetic semigroup on a graph generalizes the Jukes-Cantor binary model on a tree. Minimal generating sets of phylogenetic semigroups have been described for trivalent trees by Buczy\'nska and Wi\'sniewski, and for trivalent graphs…

In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of…

In this paper we prove that the Torelli group of a surface of genus at least 3 with 2 boundary components is finitely generated. As a consequence, we answer Putman's question on the finite generation of the stabilizer subgroup of the…

Let us denote elements of the symmetric group $S_n$ using square brackets for the one-line notation. Cycles will be represented using parentheses, following the standard cycle notation. Under this convention, the full reversal of the…

It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.

Let $F$ be a field. We show that the largest irredundant generating sets for the algebra of $n\times n $ matrices over $F$ have $2n-1$ elements when $n>1$. (A result of Laffey states that the answer is $2n-2$ when $n>2$, but its proof…

We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q) and H(2n,q^2), in…

The abelianization of the level $d$ principal congruence subgroup $\Gamma_d(n)$ of $\textrm{SL}(n;\mathbb{Z})$ was determined by Lee-Szczarba. By this result and a result of Tits, we can obtain a minimal generating set for $\Gamma_d(n)$. In…

Let $N_{g,s}$ denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group $M(N_{g,s})$ of the surface $N_{g,s}$, where…

The trace algebra C_{nd} is generated by all traces of products of d generic n x n matrices. Minimal generating sets of C_{nd} and their defining relations are known for n < 3 and n = 3, d=2. This paper states a minimal generating set and…

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this…

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group \Gamma which admits the congruence…

We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We…

It is shown that the canonical ring of a minimal surface of general type with $p_g=0, K^2\geq 2$ is generated by its elements of degree lesser or equal to 5, provided $|2K|$ has no fixed components, and that this bound can be lowered to 4…

A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such…