Semidirect product of cyclic group of prime-square order and cyclic group of prime order
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
Definition
Let 
be an odd prime. This group is, up to isomorphism, the only non-abelian group of order 
and exponent 
.
Note that for 
, these descriptions work, and yield dihedral group:D8 -- however, that group has a very different qualitative behavior from the odd case.
Notation
This group is sometimes denoted as 
or as 
.
As a Sylow subgroup of a holomorph
This group is isomorphic to the -Sylow subgroup of the holomorph of the cyclic group of order .
As a presentation
In non-abelian language, it is a group given by the presentation:
Here 
is the generator of a cyclic normal subgroup of order and 
is the generator of the complement of order acting on it.
As a semidirect product
Given a prime , this group is defined as the semidirect product of the cyclic group 
with the cyclic group 
, where 
acts on 
as follows:
More explicitly, it is the group of ordered pairs 
with 
and 
with multiplication given by:
Related groups
For any prime , there are two non-Abelian groups of order : this one and the group of upper triangular unipotent matrices over the prime field.
Arithmetic functions
Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions
For some of these, the function values are different when and/or when 
. These are clearly indicated below.
Arithmetic functions taking values between 0 and 3
| Function | Value | Explanation |
|---|---|---|
| prime-base logarithm of order | 3 | the order is
|
| prime-base logarithm of exponent | 2 | the exponent is .
|
| nilpotency class | 2 | |
| derived length | 2 | |
| Frattini length | 2 | |
| minimum size of generating set | 2 | |
| subgroup rank | 2 | |
| rank as p-group | 2 | |
| normal rank as p-group | 2 | |
| characteristic rank as p-group | 2 | the subgroup comprising the identity element and the elements of order is a characteristic elementary abelian subgroup of order .
|
Arithmetic functions of a counting nature
| Function | Value | Explanation |
|---|---|---|
| number of conjugacy classes |  |
elements in the center, and each other conjugacy class has size
|
| number of subgroups |  when  ,  when |
|
| number of normal subgroups |  |
|
| number of conjugacy classes of subgroups |  for ,  for |
Subgroups
Further information: Subgroup structure of semidirect product of cyclic group of prime-square order and cyclic group of prime order
The subgroups are as follows:
- The trivial subgroup. (1)
- The center, which is the subgroup
, or the multiples of in the cyclic group. It is also the commutator subgroup, the Frattini subgroup and the socle. (That all these are the same indicates that this group is an extraspecial group). Isomorphic to group of prime order. (1)
- Subgroups of order generated by conjugates of . They form a single conjugacy class of size . ()
- The subgroup of order generated by
and : in other words, the multiples of in the cyclic normal subgroup, and the element of order acting on it. This is a fully invariant subgroup. Isomorphic to elementary abelian group of prime-square order. (1)
- Cyclic normal subgroups of order , generated by elements of the form
. All these are automorphic to the cyclic subgroup 
, though each one is normal, so no two of them are conjugate. Isomorphic to cyclic group of prime-square order. ()
- The whole group. (1)
Types (1), (2), (4) and (6) are characteristic, while type (5) is normal not not characteristic. Type (3) comprises automorph-conjugate subgroups that are not normal.
Endomorphisms
Inner automorphisms
The inner automorphisms preserve the normal subgroup , and they act via multiplication of every element in it by 
for 
in .
Outer automorphisms
These automorphisms permute the subgroups of order . Here are some examples of outer automorphisms:
where 
is an invertible element modulo .
Note that no automorphism can take 
to 
for distinct and 
. The retract comprising s is thus a quasicharacteristic retract. Since the automorphism group of the kernel is an Abelian group, we in fact conclude that the retract is a relatively rigid subgroup.
