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 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.

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].

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.

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.

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.

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〉.

scholarly journals On Schur p-Groups of odd order

2017 ◽  
Vol 16 (03) ◽  
pp. 1750045 ◽  
Author(s):  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Odd Order ◽  
A Finite Group ◽  
Natural Way

A finite group [Formula: see text] is called a Schur group if any [Formula: see text]-ring over [Formula: see text] is associated in a natural way with a subgroup of [Formula: see text] that contains all right translations. We prove that the groups [Formula: see text], where [Formula: see text], are Schur. Modulo previously obtained results, it follows that every noncyclic Schur [Formula: see text]-group, where [Formula: see text] is an odd prime, is isomorphic to [Formula: see text] or [Formula: see text], [Formula: see text].

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.

Subgroups of Central Separable Algebras

1973 ◽  
Vol 25 (4) ◽  
pp. 881-887 ◽  
Author(s):  
E. D. Elgethun
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Division Ring ◽  
Multiplicative Group ◽  
Prime Field ◽  
Odd Order ◽  
Multiplicative Subgroup ◽  
A Finite Group ◽  
Separable Algebras

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

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]

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