A CHARACTERISATION OF CERTAIN FINITE GROUPS OF ODD ORDER

2011 ◽  
Vol 111 (-1) ◽  
pp. 67-76
Author(s):  
Ashish Kumar Das ◽  
Rajat Kanti Nath
Keyword(s):  
Finite Groups ◽  
Odd Order

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 centralizers of elements of odd order in finite groups

1979 ◽  
Vol 61 (1) ◽  
pp. 269-280 ◽  
Author(s):  
Zvi Arad ◽  
David Chillag
Keyword(s):  
Finite Groups ◽  
Odd Order

A CHARACTERISATION OF CERTAIN FINITE GROUPS OF ODD ORDER

2011 ◽  
Vol 111A (2) ◽  
pp. 67-76
Author(s):  
Ashish Kumar Das ◽  
Rajat Kanti Nath
Keyword(s):  
Finite Groups ◽  
Odd Order

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 Finite Groups Which Admit An Automorphism With Few Orbits

1960 ◽  
Vol 12 ◽  
pp. 73-100 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Groups ◽  
Study Groups ◽  
Quaternion Group ◽  
Fixed Integer ◽  
Fixed Element ◽  
Odd Order ◽  
Special Case

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

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 Word problems for finite nilpotent groups

2020 ◽  
Vol 115 (6) ◽  
pp. 599-609
Author(s):  
Rachel D. Camina ◽  
Ainhoa Iñiguez ◽  
Anitha Thillaisundaram
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Character Degrees ◽  
Character Theory ◽  
Odd Order ◽  
Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

Finite groups with Abelian centralizers of elements of odd order

1977 ◽  
Vol 16 (4) ◽  
pp. 255-260 ◽  
Author(s):  
V. M. Busarkin
Keyword(s):  
Finite Groups ◽  
Odd Order

On groups satisfying the converse of Lagrange's theorem

1974 ◽  
Vol 75 (1) ◽  
pp. 25-32 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Factor Group ◽  
Proved Theorem ◽  
Supersoluble Groups ◽  
Odd Order ◽  
Sylow Tower ◽  
Rank Of A Group ◽  
Lagrange's Theorem

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

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