Nilpotency of solvable groups admitting a splitting automorphism of prime order

1980 ◽  
Vol 19 (1) ◽  
pp. 77-84 ◽  
Author(s):  
E. I. Khukhro

1959 ◽  
Vol 11 ◽  
pp. 311-320 ◽  
Author(s):  
Daniel Gorenstein ◽  
I. N. Herstein

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 In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable.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.









2016 ◽  
Vol 15 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Hadi Ahmadi ◽  
Bijan Taeri

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.







2013 ◽  
Vol 11 (9) ◽  
Author(s):  
James Beidleman ◽  
Hermann Heineken ◽  
Jack Schmidt

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.



1992 ◽  
Vol 34 (2) ◽  
pp. 360-362
Author(s):  
E. I. Khukhro


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