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 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.

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

Some results on the classification of expansive automorphisms of compact abelian groups

1996 ◽  
Vol 16 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Fabio Fagnani
Keyword(s):  
Finite Group ◽  
Topological Entropy ◽  
Abelian Groups ◽  
Topological Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Abelian Groups ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

scholarly journals Duality for finite abelian hypergroups over splitting fields

1979 ◽  
Vol 20 (1) ◽  
pp. 57-70 ◽  
Author(s):  
J.R. McMullen ◽  
J.F. Price
Keyword(s):  
Finite Group ◽  
Duality Theory ◽  
Abelian Groups ◽  
Conjugacy Classes ◽  
Finite Abelian Groups ◽  
Precise Sense ◽  
Classical Duality ◽  
A Finite Group

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals Groups fixing graphas in switching classes

1982 ◽  
Vol 33 (1) ◽  
pp. 76-85
Author(s):  
Martin W. Liebeck
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Finite Set ◽  
A Finite Group ◽  
Switching Class

AbstractA permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 541-546
Author(s):  
Jiangtao Shi ◽  
Klavdija Kutnar ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

An answer to a conjecture on the sum of element orders

2020 ◽  
pp. 2250067
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi ◽  
Morteza Jafarpour
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Nilpotent Group ◽  
Prime Number ◽  
Lower Bounds ◽  
Finite Groups ◽  
Open Problem ◽  
Element Orders ◽  
Equality Condition ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

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