Nilpotency of solvable groups admitting a splitting automorphism of prime order

1980 ◽  
Vol 19 (1) ◽  
pp. 77-84 ◽  
Author(s):  
E. I. Khukhro
Keyword(s):  
Prime Order ◽  
Solvable Groups ◽  
Splitting Automorphism

scholarly journals A Class of Solvable Groups

1959 ◽  
Vol 11 ◽  
pp. 311-320 ◽  
Author(s):  
Daniel Gorenstein ◽  
I. N. Herstein
Keyword(s):  
Finite Groups ◽  
Prime Order ◽  
Solvable Groups ◽  
Unpublished Paper

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.

scholarly journals Automorphism of prime order of solvable groups whose subgroups of fixed points are nilpotent

1984 ◽  
Vol 88 (1) ◽  
pp. 178-189 ◽  
Author(s):  
A.O Asar
Keyword(s):  
Fixed Points ◽  
Prime Order ◽  
Solvable Groups

scholarly journals Fitting height of solvable groups admitting an automorphism of prime order with abelian fixed-point subgroup

1981 ◽  
Vol 68 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Arnold D Feldman
Keyword(s):  
Fixed Point ◽  
Prime Order ◽  
Solvable Groups ◽  
Fitting Height

A solvable group admitting a regular splitting automorphism of prime order is nilpotent

1978 ◽  
Vol 17 (5) ◽  
pp. 402-406 ◽  
Author(s):  
E. I. Khukhro
Keyword(s):  
Solvable Group ◽  
Prime Order ◽  
Regular Splitting ◽  
Splitting Automorphism

Planarity of the intersection graph of subgroups of a finite group

2016 ◽  
Vol 15 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Hadi Ahmadi ◽  
Bijan Taeri
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Prime Order ◽  
Undirected Graph ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
A Finite Group

For a nontrivial finite group [Formula: see text] different from a cyclic group of prime order, the intersection graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are the nontrivial proper subgroups of [Formula: see text] and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.

scholarly journals Solvable groups admitting an “almost fixed point free” automorphism of prime order

1978 ◽  
Vol 22 (2) ◽  
pp. 191-207 ◽  
Author(s):  
Stephen M. Gagola
Keyword(s):  
Fixed Point ◽  
Prime Order ◽  
Solvable Groups

scholarly journals Solvable groups having system normalizers of prime order

1973 ◽  
Vol 183 ◽  
pp. 165-165
Author(s):  
Gary M. Seitz
Keyword(s):  
Prime Order ◽  
Solvable Groups

On a class of finite solvable groups

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
James Beidleman ◽  
Hermann Heineken ◽  
Jack Schmidt
Keyword(s):  
Solvable Group ◽  
Prime Order ◽  
Solvable Groups ◽  
Finite Solvable Group ◽  
Normal Subgroups ◽  
Nilpotent Residual ◽  
Quotient Groups ◽  
Subnormal Subgroups

AbstractA finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.

On groups with a splitting automorphism of prime order

1992 ◽  
Vol 34 (2) ◽  
pp. 360-362
Author(s):  
E. I. Khukhro
Keyword(s):  
Prime Order ◽  
Splitting Automorphism
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