Non-CLT groups of order pq 3

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Marius Tărnăuceanu
Keyword(s):  
Finite Groups ◽  
Lagrange's Theorem

AbstractIn this note we give a characterization of finite groups of order pq 3 (p, q primes) that fail to satisfy the Converse of Lagrange’s Theorem.

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

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).

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.

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