Subgroup structure of semidirect product of cyclic group of prime-square order and cyclic group of prime order

This article gives specific information, namely, subgroup structure, about a family of groups, namely: semidirect product of cyclic group of prime-square order and cyclic group of prime order.
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Let $p$ be an odd prime.

This article discusses in detail the subgroup structure of the unique (up to isomorphism) non-abelian group of order $p^{3}$ and exponent $p^{2}$.

Note that although the definition of the group makes sense for $p=2$, yielding dihedral group:D8, the qualitative behavior for $p=2$ is very different.

We use here the presentation:

$G:=\langle g,h\mid g^{p^{2}}=h^{p}=e,gh=hg^{p+1}\rangle$.

The subgroups are as follows:

1. The trivial subgroup. (1)
2. The center, which is the subgroup $\langle g^{p}\rangle$, or the multiples of $p$ 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)
3. Subgroups of order $p$ generated by conjugates of $h$. They form a single conjugacy class of size $p$. ($p$)
4. The subgroup of order $p^{2}$ generated by $g^{p}$ and $h$: in other words, the multiples of $p$ in the cyclic normal subgroup, and the element of order $p$ acting on it. This is a fully invariant subgroup. Isomorphic to elementary abelian group of prime-square order. (1)
5. Cyclic normal subgroups of order $p^{2}$, generated by elements of the form $gh^{k}$. All these are automorphic to the cyclic subgroup $\langle g\rangle$, though each one is normal, so no two of them are conjugate. Isomorphic to cyclic group of prime-square order. ($p$)
6. The whole group. (1)

We first give a quick summary:

• Except the subgroups in (3), all subgroups are normal. The subgroups in (3) form exactly one conjugacy class of size $p$, all of which live in the group of type (4).
• The subgroups of type (1), (2), (4) and (6) are characteristic, and in fact are all fully invariant.

Tables for quick information

Table classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1
Group of prime order	$(p,1)$	$p+1$	2	1	1
Cyclic group of prime-square order	$(p^{2},1)$	$p$	$p$	$p$	0
Elementary abelian group of prime-square order	$(p^{2},2)$	1	1	1	1
Whole group	$(p^{3},4)$	1	1	1	1
Total	--	$2p+4$	$p+5$	$p+4$	4

Table listing number of subgroups by order

Group name	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
$1$	1	1	1	1
$p$	$p+1$	2	1	1
$p^{2}$	$p+1$	$p+1$	$p+1$	1
$p^{3}$	1	1	1	1
Total	$2p+4$	$p+5$	$p+4$	4

The center (type (2))

This is a $p$-element subgroup generated by $g^{p}$.

Subgroup-defining functions yielding this subgroup

Note that equality of the first four is essentially the fact that the group is an extraspecial group.

• The center
• The commutator subgroup
• The Frattini subgroup
• The socle
• The first agemo subgroup
• The ZJ-subgroup: The join of abelian subgroups of maximum order of the whole group is the whole group, so the ZJ-subgroup of the whole group equals its center.

Subgroup properties satisfied by this subgroup

On account of being an agemo subgroup as well as on account of being the commutator subgroup, this subgroup is a verbal subgroup -- it is a subgroup generated by words of a certain form. Thus, it satisfies the following properties:

• Fully invariant subgroup: For full proof, refer: Verbal implies fully invariant
• Image-closed fully invariant subgroup: For full proof, refer: Verbal implies image-closed fully invariant
• Characteristic subgroup
• Image-closed characteristic subgroup

It also satisfies the following subgroup properties for obvious reasons:

• Central subgroup, central factor
• Simple normal subgroup, transitively normal subgroup

Subgroup properties not satisfied by this subgroup

• Homomorph-containing subgroup
• Isomorph-free subgroup
• Intermediately characteristic subgroup: For instance, it is not characteristic in the subgroup of type (4), the elementary abelian subgroup of order $p^{2}$.
• Complemented normal subgroup, Lattice-complemented subgroup, Permutably complemented subgroup: There is no complement to the center. This is a phenomenon for all nilpotent groups, and follows from the fact that nilpotent implies center is normality-large.

The elementary abelian subgroup of prime-square order (type (4))

Subgroup-defining functions yielding this subgroup

• first omega subgroup: In other words, it is the subgroup generated by all the elements of order $p$. Note that since omega-1 of odd-order class two p-group has prime exponent, every non-identity element in this group has order $p$. In fact, in this case, it is elementary abelian of order $p^{2}$.
• Baer norm: This subgroup equals the intersection of the normalizers of all subgroups.
• Wielandt subgroup: This subgroup equals the intersection of the normalizers of all subnormal subgroups.

Subgroup properties satisfied by this subgroup

• Large subgroup: The intersection with any nontrivial subgroup is nontrivial. This follows from its being an omega subgroup.
• Homomorph-containing subgroup: All omega subgroups are homomorph-containing.
• Fully invariant subgroup: All omega subgroups are fully invariant.
• Isomorph-free subgroup: There is no other subgroup isomorphic to this subgroup. This is again true for omega subgroups in general.
• Intermediately characteristic subgroup
• Maximal characteristic subgroup
• Maximal normal subgroup
• Maximal among abelian characteristic subgroups
• Maximal among abelian normal subgroups
• Self-centralizing subgroup: This follows, for instance, from the fact that maximal among abelian normal implies self-centralizing.
• Coprime automorphism-faithful subgroup: This follows from the fact that normal and self-centralizing implies coprime automorphism-faithful. Alternatively, it follows from the fact that omega-1 of odd-order p-group is coprime automorphism-faithful.
• Normality-large subgroup: This follows from the fact that normal and self-centralizing implies normality-large, and it also follows from the fact that large implies normality-large.
• Critical subgroup

Subgroup properties not satisfied by this subgroup

• Verbal subgroup: This subgroup cannot be expressed as the subgroup generated by all elements that can be expressed using words of a certain form.
• Image-closed fully invariant subgroup: There exist surjective homomorphisms under which the image of this subgroup is not fully invariant.
• Central factor
• Transitively normal subgroup: There exist normal subgroups of this subgroup that are not normal in the whole group.
• Complemented normal subgroup: Any element outside this subgroup has order $p^{2}$, hence this subgroup is not complemented. This is again general to the proper nontrivial omega subgroups -- they are large, and hence, not complemented.

The conjugacy class of non-normal subgroups of order $p$ (type (3))

Subgroup properties satisfied by these subgroups

• Automorph-conjugate subgroup: This conjugacy class of subgroups is preserved under all automorphisms of the whole group.
• Subgroup contained in the Baer norm
• 2-subnormal subgroup: This follows from its being contained in the Baer norm.
• Permutable subgroup: This also follows from its being contained in the Baer norm.
• Retract: Each of these subgroups has a normal complement: a cyclic group of order $p^{2}$.
• 2-hypernormalized subgroup

Subgroup properties not satisfied by these subgroups

• Characteristically complemented subgroup
• Normal subgroup

Cyclic subgroups of prime-square order (Type (5))

Subgroup properties satisfied by these subgroups

• Maximal among abelian normal subgroups
• Cyclic normal subgroup
• Cyclic maximal subgroup
• Complemented normal subgroup
• Transitively normal subgroup
• Self-centralizing subgroup: This follows, for instance, from the fact that maximal among abelian normal implies self-centralizing in nilpotent
• Coprime automorphism-faithful subgroup: This follows, for instance, from the fact that Normal and self-centralizing implies coprime automorphism-faithful.
• Isomorph-automorphic subgroup

Subgroup properties not satisfied by these subgroups

• Central factor
• Automorph-conjugate subgroup, characteristic subgroup