scholarly journals On 10-Centralizer Groups of Odd Order

ISRN Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Z. Foruzanfar ◽  
Z. Mostaghim
Keyword(s):  
Finite Groups ◽  
Nonabelian Group ◽  
Odd Order

Let G be a group, and let Cent(G) denote the number of distinct centralizers of its elements. A group G is called n-centralizer if Cent(G)=n. In this paper, we investigate the structure of finite groups of odd order with Cent(G)=10 and prove that there is no finite nonabelian group of odd order with Cent(G)=10.

scholarly journals Small squaring and cubing properties for finite groups

1991 ◽  
Vol 44 (3) ◽  
pp. 429-450 ◽  
Author(s):  
Ya. G. Berkovich ◽  
G.A. Freiman ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Groups ◽  
Classification Problem ◽  
Nonabelian Group ◽  
Derived Length ◽  
Odd Order ◽  
Index 2

A group G is said to have the small squaring property on k-sets if |K2| < k2 for all k-element subsets K of G, and is said to have the small cubing property on k-sets if |K3| < k3 for all k-element subsets K. It is shown that a finite nonabelian group with the small squaring property on 3-sets is either a 2-group or is of the form TP with T a normal abelian odd order subgroup and P a nontrivial 2-group such that Q = Cp(T) has index 2 in P and P inverts T. Moreover either P is abelian and Q is elementary abelian, or Q is abelian and each element of P − Q inverts Q. Conversely each group of the form TP as above has the small squaring property on 3-sets. As for the nonabelian 2-groups with the small squaring property on 3-sets, those of exponent greater then 4 are classified and the examples are similar to dihedral or generalised quaternion groups. The remaining classification problem of exponent 4 nonabelian examples is not complete, but these examples are shown to have derived length 2, centre of exponent at most 4, and derived quotient of exponent at most 4. Further it is shown that a nonabelian group G satisfies |K2| < 7 for all 3-element subsets K if and only if G = S3. Also groups with the small cubing property on 2-sets are investigated.

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.

Suatu Hubungan Kumpulan Pusat–2 dengan Teori Kebarangkalian

2012 ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Hasimah Sapiri
Keyword(s):  
Probability Theory ◽  
Finite Group ◽  
Finite Groups ◽  
Random Element ◽  
Symmetric Groups ◽  
Finite Rings ◽  
Nonabelian Group ◽  
Central Group ◽  
Upper Limit

Penentuan darjah keabelanan bagi suatu kumpulan tak abelan telah diperkenalkan untuk kumpulan simetri oleh Erdos dan Turan [1]. Dalam tahun 1973, Gustafson [2] mengkajinya bagi kumpulan terhingga sementara MacHale [3] mengkajinya bagi gelanggang terhingga dalam tahun 1976. Dalam kajian ini, beberapa keputusan yang berkaitan dengan Pn(G), kebarangkalian bahawa suatu unsur rawak dengan kuasa ke–n dalam suatu kumpulan pusat–2 G adalah kalis tukar tertib dengan unsur rawak yang lain dalam kumpulan yang sama, akan diberikan. Seterusnya, batas atas bagi P2(G) diperoleh. Kata kunci: Teori kebarangkalian, teori kumpulan, kumpulan terhingga, kalis tukar tertib The determination of the abelianness of a nonabelian group has been introduced for symmetric groups by Erdos & Turan [1]. In 1973, Gustafson [2] did this research for the finite groups while MacHale [3] determined the abelianness for finite rings in 1976. In this research, some results on Pn(G), the probability that the n–th power of a random element in a 2–central group G commutes with another random element from the same group, will be presented. Furthermore, the upper limit of P2(G) is obtained. Key words: Probability theory, group theory, finite group, commutative

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.

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