scholarly journals The characterization of finite groups whose Sylow 2-subgroups are of type L3(q), q even

1973 ◽  
Vol 25 (3) ◽  
pp. 490-512 ◽  
Author(s):  
Michael J Collins
Keyword(s):  
Finite Groups

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals Thompson-like characterization of solubility for products of finite groups

2020 ◽  
Author(s):  
P. Hauck ◽  
L. S. Kazarin ◽  
A. Martínez-Pastor ◽  
M. D. Pérez-Ramos
Keyword(s):  
Finite Groups

scholarly journals Finitep′-nilpotent groups. II

1987 ◽  
Vol 10 (2) ◽  
pp. 303-309 ◽  
Author(s):  
S. Srinivasan
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Complete Characterization

In this paper we continue the study of finitep′-nilpotent groups that was started in the first part of this paper. Here we give a complete characterization of all finite groups that are notp′-nilpotent but all of whose proper subgroups arep′-nilpotent.

scholarly journals The characterization of finite groups with dihedral Sylow 2-subgroups. III

1965 ◽  
Vol 2 (3) ◽  
pp. 354-393 ◽  
Author(s):  
Daniel Gorenstein ◽  
John H Walter
Keyword(s):  
Finite Groups

Finite groups with given sets of 𝔉-subnormal subgroups

2019 ◽  
Vol 13 (04) ◽  
pp. 2050073 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Groups ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Hereditary Saturated Formation ◽  
Subnormal Subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

scholarly journals On solvable groups with one vanishing class size

2020 ◽  
pp. 1-20
Author(s):  
M. Bianchi ◽  
E. Pacifici ◽  
R. D. Camina ◽  
Mark L. Lewis
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Class Size ◽  
Single Element ◽  
Conjugacy Class Sizes ◽  
Conjugacy Class Size ◽  
Odd Order ◽  
Vanishing Elements ◽  
A Finite Group

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

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