scholarly journals Finite groups with a self-centralizing subgroup of order 4

1967 ◽  
Vol 7 (4) ◽  
pp. 570-576 ◽  
Author(s):  
Warren J. Wong
Keyword(s):  
Normal Subgroup ◽  
Early Result ◽  
Finite Groups ◽  
Present Author ◽  
Permutation Groups ◽  
Odd Order ◽  
Nontrivial Normal Subgroup

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.

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 Bounds on finite quasiprimitive permutation groups

2001 ◽  
Vol 71 (2) ◽  
pp. 243-258 ◽  
Author(s):  
Cheryl E. Praeger ◽  
Aner Shalev
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Minimum Degree ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Base Size ◽  
Nontrivial Normal Subgroup

AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

scholarly journals Translation planes of odd order and odd dimension

1979 ◽  
Vol 2 (2) ◽  
pp. 187-208 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Translation Planes ◽  
Solvable Subgroup ◽  
Joint Paper ◽  
Desarguesian Planes ◽  
Odd Order ◽  
Collineation Groups ◽  
Zero Vector

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension2d overGF(q), whereqanddare odd. If the stabilizer of the zero vector is non-solvable, letG0be a minimal normal non-solvable subgroup. We suspect thatG0must be isomorphic to someSL(2,u)or homomorphic toA6orA7. Our main result is that this is the case whendis the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow2-groups whendandqare both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order27(i.e.,dandqare both equal to3) which admitsSL(2,13).

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

The Perchloric Method for the Determination of Sulfur in Rubber

1934 ◽  
Vol 7 (4) ◽  
pp. 730-735
Author(s):  
Ernest Kahane
Keyword(s):  
Nitric Acid ◽  
Perchloric Acid ◽  
Organic Substance ◽  
Present Author ◽  
General Idea ◽  
Organic Substances ◽  
Gutta Percha ◽  
Higher Temperature ◽  
As If

Abstract The problem of the determination of sulfur in rubber has been dealt with extensively in the literature, and it seems as if discussions and descriptions of new technic are nowhere nearly ended yet. The determination is so essential, and its rapid and precise execution is of such importance in industrial technic, that efforts in this direction should not be regarded as wasted. In 1926 and in 1927 Le Caoutchouc et La Gutta-Percha contained two articles in which the present author discussed the conditions of the determination of sulfur in rubber and then proposed the use of a new oxidizing mixture, not mentioned previous to that time, which involved the destruction of organic substances by perchloric acid. This method consisted simply in the attack on a 1-gram sample of rubber by 10 cc. of nitric acid (d. 1.39) and 5 cc. of perchloric acid (d. 1.61). Upon heating, attack by the nitric acid takes place, and this is followed by evaporation of the excess nitric acid, then at a little higher temperature there is an attack by the perchloric acid, which oxidizes the rest of the organic substance completely. This publication was concerned much more, in the determination of sulfur by the perchloric method, with the general idea of the destruction of organic substances than it was with the precise details of carrying it out. The technic had been studied somewhat superficially, as is shown by the text of the article itself.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

scholarly journals On the determination of the ramification index in Clifford's theorem

1988 ◽  
Vol 31 (3) ◽  
pp. 469-474
Author(s):  
Robert W. van der Waall
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Ramification Index ◽  
A Finite Group

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.

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

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

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