48 research outputs found
Groups whose prime graphs have no triangles
Let G be a finite group and let cd(G) be the set of all complex irreducible
character degrees of G Let \rho(G) be the set of all primes which divide some
character degree of G. The prime graph \Delta(G) attached to G is a graph whose
vertex set is \rho(G) and there is an edge between two distinct primes u and v
if and only if the product uv divides some character degree of G. In this
paper, we show that if G is a finite group whose prime graph \Delta(G) has no
triangles, then \Delta(G) has at most 5 vertices. We also obtain a
classification of all finite graphs with 5 vertices and having no triangles
which can occur as prime graphs of some finite groups. Finally, we show that
the prime graph of a finite group can never be a cycle nor a tree with at least
5 vertices.Comment: 13 page
Groups with normal restriction property
Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if,
whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of
K in G. Using the Classification of Finite Simple Groups, we prove that if
every maximal subgroup of G is an NR -subgroup then G is solvable. This gives a
positive answer to a conjecture posed in Berkovich (Houston J Math 24:631-638,
1998).Comment: 5 page
Character degree sums in finite nonsolvable groups
Let N be a minimal normal nonabelian subgroup of a finite group G. We will
show that there exists a nontrivial irreducible character of N of degree at
least 5 which is extendible to G. This result will be used to settle two open
questions raised by Berkovich and Mann, and Berkovich and Zhmud'.Comment: 5 page
Primitive permutation groups and derangements of prime power order
Let be a transitive permutation group on a finite set of size at least
. By a well known theorem of Fein, Kantor and Schacher, contains a
derangement of prime power order. In this paper, we study the finite primitive
permutation groups with the extremal property that the order of every
derangement is an -power, for some fixed prime . First we show that these
groups are either almost simple or affine, and we determine all the almost
simple groups with this property. We also prove that an affine group has
this property if and only if every two-point stabilizer is an -group. Here
the structure of has been extensively studied in work of Guralnick and
Wiegand on the multiplicative structure of Galois field extensions, and in
later work of Fleischmann, Lempken and Tiep on -semiregular pairs.Comment: 30 pages; to appear in Manuscripta Mat