Divisores primos del orden de productos de elementos en grupos finitos

• Autores: Azahara Sáez Porras
• Directores de la Tesis: Alexander Moretó Quintana (dir. tes.)
• Lectura: En la Universitat de València ( España ) en 2018
• Idioma: español
• Número de páginas: 74
• Tribunal Calificador de la Tesis: Silvio Dolfi (presid.) , Lucía Sanus Vitoria (secret.) , Josu Sangroniz Gómez (voc.)
• Enlaces
  • Tesis en acceso abierto en: RODERIC
• Resumen

  • A classical problem in Group Theory is the study of the relationship between the structure of a finite group and the lengths of its conjugacy classes or its characters degrees. Originally, the objective of this thesis was to obtain results on orders of elements analogous to those already existing for conjugacy classes or characters. Finally, we have obtained new results on orders of elements that did not exist in the literature for character degrees or lengths of conjugacy classes. In particular, in this dissertation we study what information can be obtained about the structure of a group from the relationship between the orders of certain elements and their products.

    In Chapter 1 we establish the notation and remember some results and general definitions of Group Theory that will be useful to us.

    In Chapter 2 we mention well-known results that involve orders of elements which give information about the structure of the group or a criterion of belonging of a given element to a relevant normal subgroup. In addition, we obtain some variants of a theorem of J. G. Thompson (and of a theorem of R. M. Guralnick and P. H. Tiep) that characterizes the solvability in terms of products of elements of coprime orders (and prime power order).

    One of the first objectives of this thesis and in particular of Chapter 3, is to obtain the version for elements of prime power order of a theorem of B. Baumslag and J. Wiegold, which asserts that a group G is nilpotent if and only if for all pairs of elements x, y of G of coprime order, we have that the order of xy is equal to the product of the order of the element x by the order of the element y. Motivated by this result, we obtain several similar ones, as well as a series of local results. In particular, we prove a criterion for the existence of nilpotent Hall subgroups in any group and a characterization of the existence of normal Hall subgroups.

    In Chapter 4 we try to obtain results in line with those of Chapter 3 but to characterize the membership of a fixed element to certain relevant subgroups. One of the main problems that we consider in this chapter is whether it is true that a p-element x of G satisfies that x belongs to the largest normal p-subgroup if and only if for all primes q other than p, q divides the order of xy for all non-trivial q-elements y. We have shown that this question has an affirmative answer for all groups that do not contain composition factors of Lie type in characteristic p and that a minimal counterexample to this question is an almost-simple group with socle one of these groups.

    Finally, in Chapter 5 we show some results analogous to those that we have seen in the central chapters but for conjugacy classes and characters.