The Frobeniusgroup of order 20
is the set of maps from to of the form
I'll answer your questions in
slightly different order:
d=1: 1 element, the identity
d=2: 5 elements,
d=4: 10 elements,
d=5: 4 elements,
A presentation for is
The elements of order 5 and two together lie in the
Dihedral group of generated by and (of order 10).
This group has trivial center. Hence has trivial center.
3) There is only one Sylow 5 subgroup, the group
generated by (of order 5).
There are Sylow 2 subgroups (of order 4).
They are the groups generated by and its conjugates
4) There are two normal subgroups, both mentioned already:
The Sylow 5 subgroup generated by ,
and the Dihedral group of order 10 generated by and