Each group considered will have a finite number of elements. A

group is solvable if there exists a sequence of subgroups such that

for each consecutive pair, the first one is a normal subgroup of the

second one, and the order (number of elements) of the second one is a

prime number times the order of the first one. For example, the group

of permutations of a set with 4 elements is of order 4! = 24, and it

is solvable because it possesses a sequence of subgroups with each

normal in the next whose orders are 1, 2, 4, 12, and 24. We say H is

a Sylow p-subgroup of G if p is a prime and the order of H is the

highest power of p that divides the order of G. In the 1870's, Sylow

proved that each finite group has a Sylow p-subgroup for each prime p

and that furthermore, all the Sylow p-subgroups of G are conjugate,

and any conjugate of a Sylow p-subgroup is a Sylow p-subgroup.

The concept of injector is one of several generalizations of the

Sylow subgroups. If F is a particular type of set of subgroups of a

solvable group G called a Fitting set of G, then F contains a

collection of subgroups which are maximal in a nice way. These

subgroups are called the F-injectors of G, and they satisfy the

conditions above mentioned for Sylow p-subgroups: for each Fitting set

F of a solvable group G, there exists a conjugacy class of subgroups

consisting of all the F-injectors of G. The Sylow p-subgroups of G

are the F-injectors of G when F is the Fitting set of p-subgroups of

G, i.e, the subgroups whose orders are powers of the prime p.

For the last several years, I have been working with Rex Dark

of the National University of Ireland, Galway, and Maria Dolores Perez-

Ramos of the University of Valencia, Spain to characterize the set of

injectors of a finite solvable group G without reference to Fitting

sets. Thus we have been discovering properties of a subgroup H that

are necessary and sufficient for the existence of some Fitting set F

for which H is an F-injector. In this talk, I will define Fitting sets

and injectors, relate those concepts to Sylow subgroups, and mention

some of our results.

Laura Ciobanu

When? | 24.05.2011 17:15 |
---|---|

Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |

Contact | Department of Mathematics isabella.schmutz@unifr.ch |