ON AN INVERSE PROBLEM TO FROBENIUS' THEOREM II

2012 ◽  
Vol 11 (05) ◽  
pp. 1250092 ◽  
Author(s):  
WEI MENG ◽  
JIANGTAO SHI ◽  
KELIN CHEN
Keyword(s):  
Inverse Problem ◽  
Finite Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Complete Classification ◽  
Frobenius Theorem ◽  
A Finite Group

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting Le(G) = {x ∈ G|xe = 1}. Frobenius proved that |Le(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||Le(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

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 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.

A CLASSIFICATION OF FINITE p-GROUPS WHOSE PROPER SUBGROUPS ARE OF CLASS ≤ 2 (II)

2012 ◽  
Vol 12 (03) ◽  
pp. 1250171 ◽  
Author(s):  
PUJIN LI
Keyword(s):  
Finite Group ◽  
Complete Classification ◽  
A Finite Group

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of minimal non-[Formula: see text]p-groups G with G3 ≅ Cp × Cp for p > 3.

A CLASSIFICATION OF FINITE p-GROUPS WHOSE PROPER SUBGROUPS ARE OF CLASS ≤ 2 (I)

2012 ◽  
Vol 12 (03) ◽  
pp. 1250170 ◽  
Author(s):  
PUJIN LI
Keyword(s):  
Finite Group ◽  
Complete Classification ◽  
Odd Order ◽  
A Finite Group

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of p-groups H of odd order with d(H) = 2 and c(H) = 2. Based on the classification of H, minimal non-[Formula: see text]p-groups G are classified for p > 3. If p > 3, then we have G3 ≅ Cp or G3 ≅ Cp × Cp. In this paper, we deal with the case when G3 ≅ Cp. In another paper [A classification of finite p-groups whose proper subgroups are of class ≤ 2(II), accepted] we deal with the case when G3 ≅ Cp × Cp.

scholarly journals Generalised Beauville groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ludo Carta ◽  
Ben T. Fairbairn
Keyword(s):  
Finite Group ◽  
Riemann Surfaces ◽  
Complete Classification ◽  
Similar Action ◽  
Compact Riemann Surfaces ◽  
A Finite Group ◽  
Beauville Group ◽  
Beauville Surface

Abstract A Beauville surface is defined by the action of a finite group (Beauville group) on the product of two compact Riemann surfaces. In this paper, we consider higher products and the possibility of a similar action by a finite group, which we call a generalised Beauville group; we prove several results regarding the existence and construction of infinite families of generalised Beauville groups and provide a complete classification of the abelian ones; we list all generalised Beauville groups of orders from 1 to 1023.

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 Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

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 Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

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