scholarly journals Groups containing locally maximal product-free sets of size 4

2021 ◽  
Vol 31 (2) ◽  
pp. 167-194
Author(s):  
C. S. Anabanti ◽  
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Open Problem ◽  
Finite Abelian Group ◽  
Locally Maximal ◽  
A Finite Group ◽  
Free Set ◽  
Free Sets

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

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.

CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH

2019 ◽  
Vol 62 (3) ◽  
pp. 544-563 ◽  
Author(s):  
HERMANN HEINEKEN ◽  
FRANCESCO G. RUSSO
Keyword(s):  
Inverse Problem ◽  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

AbstractLet k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8.

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].

Conjugate p-Subgroups of Finite Groups

1972 ◽  
Vol 24 (1) ◽  
pp. 17-28
Author(s):  
John J. Currano
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Nilpotence Class ◽  
A Finite Group

Throughout this paper, let p be a prime, P be a p-group of order pt , and ϕ be an isomorphism of a subgroup R of P of index p onto a subgroup Q which fixes no non-identity subgroup of P, setwise. In [2, Lemma 2.2], Glauberman shows that P can be embedded in a finite group G such that ϕ is effected by conjugation by some element g of G. We assume that P is thus embedded. Then Q = P ∩ Pg. Let H = 〈P,Pg〉 and V = [H,Z(Q)], so Q ⊲ H and V ⊲ H.Let E(p) be the non-abelian group of order p3 which is generated by two elements of order p. Then E(p) is dihedral if p = 2 and has exponent p if p is odd. If p is odd, then E* (p) is defined in § 2 to be a particular group of order p6 and nilpotence class three.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

A Class of Finite Groups with Abelian Sylow p-Subgroups

2006 ◽  
Vol 13 (03) ◽  
pp. 471-480
Author(s):  
Zhikai Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Defect Groups ◽  
Abelian Defect Groups ◽  
A Finite Group

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

scholarly journals Automorphisms of Finite Groups and their Fixed-Point Groups

1969 ◽  
Vol 9 (3-4) ◽  
pp. 467-477 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Order ◽  
Soluble Group ◽  
Point Groups ◽  
Derived Series ◽  
A Finite Group

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

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