scholarly journals On defining groups efficiently without using inverses

2002 ◽  
Vol 133 (1) ◽  
pp. 31-36 ◽  
Author(s):  
C. M. CAMPBELL ◽  
J. D. MITCHELL ◽  
N. RUšKUC
Keyword(s):  
Finite Group ◽  
Finite Semigroup ◽  
Group Presentation ◽  
Generating Set ◽  
Semigroup Presentation ◽  
A Finite Group

Let G be a group, and let 〈A[mid ]R〉 be a finite group presentation for G with [mid ]R[mid ][ges ][mid ]A[mid ]. Then there exists a finite semigroup presentation 〈B[mid ]Q〉 for G such that [mid ]Q[mid ]- [mid ]B[mid ] = [mid ]R[mid ]- [mid ]A[mid ]. Moreover, B is either the same generating set or else it contains one additional generator.

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 The independence graph of a finite group

2020 ◽  
Vol 193 (4) ◽  
pp. 845-856
Author(s):  
Andrea Lucchini
Keyword(s):  
Finite Group ◽  
Planar Graph ◽  
Generating Set ◽  
Delta G ◽  
A Finite Group ◽  
Minimal Generating Set

Abstract Given a finite group G, we denote by $$\Delta (G)$$ Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that $$\Delta (G)$$ Δ ( G ) is connected and classify the groups G for which $$\Delta (G)$$ Δ ( G ) is a planar graph.

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 Commutators and abelian groups

1977 ◽  
Vol 24 (1) ◽  
pp. 79-91 ◽  
Author(s):  
D. M. Rodney
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Abelian Groups ◽  
Minimal Cardinality ◽  
Generating Set ◽  
Prove Theorem ◽  
A Finite Group

AbstractIf G is a group, then K(G) is the set of commutators of elements of G. C is the class of groups such that G′ = K(G) is the minimal cardinality of any generating set of dG. We prove: Theorem A. Let G be a nilpotent group of class two such that G' is finite and d(G′) < 4.Then G < G.Theorm B. Let G be a finite group such that G′ is elementary abelian of order p3. Then G ∈ C.Theorem C. Let G be a finite group with an elementary abelian Sylow p-subgroup S, of order p2, such that S ⊆ K(G). Then S ⊆K(G).

scholarly journals Cayley Graphs of Order 27p Are Hamiltonian

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Ebrahim Ghaderpour ◽  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
Generating Set ◽  
A Finite Group

Suppose that G is a finite group, such that |G|=27p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay (G;S).

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.

The k-Nacci Sequences in Some Special Modular Groups

2014 ◽  
Vol 21 (01) ◽  
pp. 59-66 ◽  
Author(s):  
Ömür Deveci ◽  
Colin M. Campbell
Keyword(s):  
Finite Group ◽  
Seed Set ◽  
Fibonacci Sequence ◽  
Generating Set ◽  
Modular Groups ◽  
Initial Seed ◽  
A Finite Group ◽  
Made In

A k-nacci (k-step Fibonacci) sequence in a finite group is a sequence of group elements x0, x1, x2, …, xn, … for which, given an initial (seed) set x0, x1, x2, …, xj-1, each element is defined by [Formula: see text] From the definition, it is clear that the period of the k-nacci sequence in a group depends on the chosen generating set and the order in which the assignments of x0, x1, x2, …, xj-1 are made. In this paper we examine the periods of the k-nacci sequences in the groups 𝔐2, [Formula: see text] and ℜ2, where each term of the sequence is reduced modulo 2.

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.

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