Conditions of convergence of a random walk on a finite group

2021 ◽  
Author(s):  
A. L. Vyshnevetskiy
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
A Finite Group

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 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 $G$.

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

1984 ◽  
Vol 16 (03) ◽  
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 A 4, S 4, and A 5 are included.

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.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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