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 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 $G$.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

scholarly journals Splitting Groups with Basis Property

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Abdullah Aljouiee
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Special Kind ◽  
Basis Property ◽  
Generating Set ◽  
A Finite Group ◽  
Minimal Generating Set

A finite group G is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group G is said to have the basis property if, for each subgroup H≤G, H has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non-p-group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any p-group has basis property. I demonstrate some types of p-groups which are splitting as well.

scholarly journals The Generalized Order-kLucas Sequences in Finite Groups

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Ömür Deveci ◽  
Erdal Karaduman
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Generating Set ◽  
Lucas Sequences ◽  
A Finite Group ◽  
Generalized Order

We study the generalized order-kLucas sequences modulom. Also, we define theith generalized order-kLucas orbitlAi,{α1,α2,…,αk-1}(G) with respect to the generating setAand the constantsα1,α2, andαk-1for a finite groupG=〈A〉. Then, we obtain the lengths of the periods of theith generalized order-kLucas orbits of the binary polyhedral groups〈n,2,2〉,〈2,n,2〉,〈2,2,n〉and the polyhedral groups(n,2,2),(2,n,2),(2,2,n)for1≤i≤k.

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 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.

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