48 research outputs found

    Groups whose prime graphs have no triangles

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    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

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    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

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    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

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    Let GG be a transitive permutation group on a finite set of size at least 22. By a well known theorem of Fein, Kantor and Schacher, GG 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 rr-power, for some fixed prime rr. 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 GG has this property if and only if every two-point stabilizer is an rr-group. Here the structure of GG 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 r′r'-semiregular pairs.Comment: 30 pages; to appear in Manuscripta Mat
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