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DTSTART;VALUE=DATE:20110524T171500
DTEND;VALUE=DATE:20110524T171500
UID:5633@agenda.unifr.ch
DESCRIPTION:Each group considered will have a finite number of elements. A \ngroup is solvable if there exists a sequence of subgroups such that \nfor each consecutive pair, the first one is a normal subgroup of the \nsecond one, and the order (number of elements) of the second one is a \nprime number times the order of the first one. For example, the group \nof permutations of a set with 4 elements is of order 4! = 24, and it \nis solvable because it possesses a sequence of subgroups with each \nnormal in the next whose orders are 1, 2, 4, 12, and 24. We say H is \na Sylow p-subgroup of G if p is a prime and the order of H is the \nhighest power of p that divides the order of G. In the 1870's, Sylow \nproved that each finite group has a Sylow p-subgroup for each prime p \nand that furthermore, all the Sylow p-subgroups of G are conjugate, \nand any conjugate of a Sylow p-subgroup is a Sylow p-subgroup.\n<br />\n<br />\n The concept of injector is one of several generalizations of the \nSylow subgroups. If F is a particular type of set of subgroups of a \nsolvable group G called a Fitting set of G, then F contains a \ncollection of subgroups which are maximal in a nice way. These \nsubgroups are called the F-injectors of G, and they satisfy the \nconditions above mentioned for Sylow p-subgroups: for each Fitting set \nF of a solvable group G, there exists a conjugacy class of subgroups \nconsisting of all the F-injectors of G. The Sylow p-subgroups of G \nare the F-injectors of G when F is the Fitting set of p-subgroups of \nG, i.e, the subgroups whose orders are powers of the prime p.\n<br />\n<br />\n For the last several years, I have been working with Rex Dark \nof the National University of Ireland, Galway, and Maria Dolores Perez- \nRamos of the University of Valencia, Spain to characterize the set of \ninjectors of a finite solvable group G without reference to Fitting \nsets. Thus we have been discovering properties of a subgroup H that \nare necessary and sufficient for the existence of some Fitting set F \nfor which H is an F-injector. In this talk, I will define Fitting sets \nand injectors, relate those concepts to Sylow subgroups, and mention \nsome of our results.\n<br />\n<br />\nLaura Ciobanu
SUMMARY:Prof. Dr. Arnold D. FELDMAN (Lancaster / z.Zt. Galway): Solvable groups, Sylow subgroups and injectors
CATEGORIES:Colloque / Congrès / Forum
LOCATION:PER 08\, Phys 2.52\, Chemin du Musée 3\, 1700 Fribourg
URL;VALUE=URI:https://agenda.unifr.ch/e/fr/5633
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