scholarly journals Random walks on generating sets for finite groups

10.37236/1322 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
F. R. K. Chung ◽  
R. L. Graham
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Random Walks ◽  
Finite Groups ◽  
Mixing Time ◽  
Cartesian Product ◽  
Generating Sets ◽  
Random Elements ◽  
A Finite Group

We analyze a certain random walk on the cartesian product $G^n$ of a finite group $G$ which is often used for generating random elements from $G$. In particular, we show that the mixing time of the walk is at most $c_r n^2 \log n$ where the constant $c_r$ depends only on the order $r$ of $G$.

A Model for Random Random-Walks on Finite Groups

1997 ◽  
Vol 6 (1) ◽  
pp. 49-56 ◽  
Author(s):  
ANDREW S. GREENHALGH
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Random Walks ◽  
Finite Groups ◽  
Generating Set ◽  
A Finite Group ◽  
Almost All

A model for a random random-walk on a finite group is developed where the group elements that generate the random-walk are chosen uniformly and with replacement from the group. When the group is the d-cube Zd2, it is shown that if the generating set is size k then as d → ∞ with k − d → ∞ almost all of the random-walks converge to uniform in k ln (k/(k − d))/4+ρk steps, where ρ is any constant satisfying ρ > −ln (ln 2)/4.

scholarly journals Generating Random Elements in Finite Groups

10.37236/818 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Finite Group ◽  
Probability Distribution ◽  
Finite Groups ◽  
Random Elements ◽  
Random Generators ◽  
A Finite Group

Let $G$ be a finite group of order $g$. A probability distribution $Z$ on $G\ $is called $\varepsilon$-uniform if $\left\vert Z(x)-1/g\right\vert \leq\varepsilon/g$ for each $x\in G$. If $x_{1},x_{2},\dots,x_{m}$ is a list of elements of $G$, then the random cube $Z_{m}:=Cube(x_{1},\dots,x_{m}) $ is the probability distribution where $Z_{m}(y)$ is proportional to the number of ways in which $y$ can be written as a product $x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\cdots x_{m}^{\varepsilon_{m}}$ with each $\varepsilon _{i}=0$ or $1$. Let $x_{1},\dots,x_{d} $ be a list of generators for $G$ and consider a sequence of cubes $W_{k}:=Cube(x_{k}^{-1},\dots,x_{1}^{-1},x_{1},\dots,x_{k})$ where, for $k>d$, $x_{k}$ is chosen at random from $W_{k-1}$. Then we prove that for each $\delta>0$ there is a constant $K_{\delta}>0$ independent of $G$ such that, with probability at least $1-\delta$, the distribution $W_{m}$ is $1/4$-uniform when $m\geq d+K_{\delta }\lg\left\vert G\right\vert $. This justifies a proposed algorithm of Gene Cooperman for constructing random generators for groups. We also consider modifications of this algorithm which may be more suitable in practice.

scholarly journals Bounding the maximal size of independent generating sets of finite groups

2020 ◽  
pp. 1-18
Author(s):  
Andrea Lucchini ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Generating Set ◽  
Generating Sets ◽  
Number Of Generators ◽  
Maximal Size ◽  
A Finite Group ◽  
Minimal Generating Set

Abstract Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by d p (G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.

scholarly journals Finite groups with a trivial Chermak–Delgado subgroup

2018 ◽  
Vol 21 (3) ◽  
pp. 449-461 ◽  
Author(s):  
Ryan McCulloch
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Extension Theorem ◽  
Cartesian Product ◽  
Direct Factor ◽  
Minimal Group ◽  
Minimal Groups ◽  
A Finite Group ◽  
Lattice Of Subgroups

Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the Chermak–Delgado subgroup of G. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p-group lattices play a major role in the author’s constructions.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

L'application des representations au calcul explicite sur certaines chaînes de Markov finies

1984 ◽  
Vol 16 (3) ◽  
pp. 656-666 ◽  
Author(s):  
Bernard Ycart
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Irreducible Representations ◽  
Permutation Groups ◽  
A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

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