The Frobeniusgroup of order 20

is the set of maps from to of the form

where

I'll answer your questions in

slightly different order:

2)

d=1: 1 element, the identity

d=2: 5 elements,

d=4: 10 elements,

d=5: 4 elements,

1)

A presentation for is

We take

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

Best,

ZD