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 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 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.

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

Blocks of defect zero in finite groups with conjugate subgroups of a given prime order

2017 ◽  
Vol 16 (11) ◽  
pp. 1750217
Author(s):  
Tianze Li ◽  
Yanjun Liu ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Groups ◽  
Prime Order ◽  
Text Block ◽  
Order Group ◽  
Odd Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. In this note, we show that if [Formula: see text] and all subgroups of [Formula: see text] of order [Formula: see text] are conjugate, then either [Formula: see text] has a [Formula: see text]-block of defect zero, or [Formula: see text] and [Formula: see text] is a direct product of a simple group [Formula: see text] and an odd order group. This improves one of our previous works.

Groups whose vanishing elements are of odd order

2018 ◽  
Vol 17 (10) ◽  
pp. 1850186
Author(s):  
S. M. Robati
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Character Formula ◽  
Odd Order ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that an element [Formula: see text] in [Formula: see text] is a vanishing element if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we study the structure of finite groups whose vanishing elements are of odd order.

scholarly journals SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

2010 ◽  
Vol 82 (2) ◽  
pp. 265-273 ◽  
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
London Math ◽  
Characteristic Property ◽  
Soluble Groups ◽  
P Elements ◽  
Odd Order ◽  
Solvability Criterion ◽  
A Finite Group

AbstractWe observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

scholarly journals Cyclic permutable subgroups of finite groups

2001 ◽  
Vol 71 (2) ◽  
pp. 169-176 ◽  
Author(s):  
John Cossey ◽  
Stewart E. Stonehewer
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Normal Closure ◽  
Permutable Subgroup ◽  
Odd Order ◽  
Permutable Subgroups ◽  
A Finite Group

AbstractThe authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.

scholarly journals POWER MAP PERMUTATIONS AND SYMMETRIC DIFFERENCES IN FINITE GROUPS

2011 ◽  
Vol 10 (05) ◽  
pp. 947-959 ◽  
Author(s):  
MÁRTON HABLICSEK ◽  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Finite Group ◽  
Large Class ◽  
Finite Groups ◽  
Difference Method ◽  
Symmetric Difference ◽  
Quadratic Reciprocity ◽  
Equivariant Maps ◽  
Odd Order ◽  
A Finite Group

Let G be a finite group. For all a ∈ ℤ, such that (a, |G|) = 1, the function ρa: G → G sending g to ga defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer dG such that [Formula: see text] for all G in a large class of groups, containing all finite nilpotent and odd order groups.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

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