24
May

## Prof. Dr. Arnold D. FELDMAN (Lancaster / z.Zt. Galway): Solvable groups, Sylow subgroups and injectors

Academic or specialist Colloquium / Congress / Forum
24.05.2011 17:15
Onsite

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 PER 08 Phys 2.52Chemin du Musée 31700 Fribourg Department of Mathematicsisabella.schmutz@unifr.ch
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