Classification of groups of prime-cube order

Statement

Let $p$ be a prime number. Then there are, up to isomorphism, five groups of order $p^3$. These include three abelian groups and two non-abelian groups. The nature of the two non-abelian groups is somewhat different for the case $p=2$.

For more information on side-by-side comparison of the groups for odd primes, see groups of prime-cube order. For information for the prime 2, see groups of order 8

The three abelian groups

The three abelian groups correspond to the three partitions of 3:

Partition of 3	Corresponding abelian group	GAP ID among groups of order $p^3$
3	cyclic group of prime-cube order, denoted $C_{p^3}$ or ${\mathbb{Z}}_{p^3}$, or $\mathbb{Z}/p^3\mathbb{Z}$	1
2 + 1	direct product of cyclic group of prime-square order and cyclic group of prime order, denoted $C_{p^2}\times C_p$ or ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$	2
1 + 1 + 1	elementary abelian group of prime-cube order, denoted $E_{p^3}$, or $C_p\times C_p\times C_p$, or ${\mathbb{Z}}_p\times {\mathbb{Z}}_p\times {\mathbb{Z}}_p$	5

The two non-abelian groups

For the case $p=2$, these are dihedral group:D8 (GAP ID: (8,3)) and quaternion group (GAP ID: (8,4)).

For the case of odd $p$, these are prime-cube order group:U(3,p) (GAP ID: ($p^3$,3)) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (GAP ID: ($p^3$,4)).

Facts used

1. Prime power order implies not centerless
2. Center is normal
3. Cyclic over central implies abelian
4. Lagrange's theorem
5. Equivalence of definitions of group of prime order: This basically states that any group of prime order must be cyclic.
6. Classification of groups of prime-square order
7. Structure theorem for finitely generated abelian groups
8. Class two implies commutator map is endomorphism
9. Formula for powers of product in group of class two

Proof

First part of proof: crude descriptions of center and quotient by center

Given: A prime number $p$, a group $P$ of order $p^3$.

To prove: Either $P$ is abelian, or we have: $Z(P)$ is a cyclic group of order $p$ and $P/Z(P)$ is an elementary abelian group of order $p^2$

Proof: Let $Z=Z(P)$ be the center of $P$.

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
CD1	$Z$ is nontrivial	Fact (1)	$P$ has order $p^3$, specifically, a power of a prime		Fact+Given direct
CD2	The order of $Z$ cannot be $p^2$	Facts (2), (3), (4), (5)	$P$ has order $p^3$		[SHOW MORE] If $Z$ has order $p^2$, then $P/Z$ (which exists by fact (2)) has order $p$ (by fact (4)) hence must be cyclic (by fact (5)). Then, by fact (3), $P$ would be abelian, but this would imply that $Z=P$, in which case the order of $Z$ would be $p^3$.
CD3	The order of $Z$ is either $p$ or $p^3$	Fact (4)	$P$ has order $p^3$	Steps (CD1), (CD2)	[SHOW MORE] By fact (4), the order of $Z$ must divide the order of $P$. The only possibilities are $1,p,p^2,p^3$. Step (CD1) eliminates the possibility of $1$, and step (CD2) eliminates the possibility of $p^2$. This leaves only $p$ or $p^3$.
CD4	If $Z$ has order $p$, then $Z$ is cyclic of order $p$ and the quotient $P/Z$ is elementary abelian of order $p^2$	Facts (3), (4), (5), (6)	$P$ has order $p^3$.		[SHOW MORE] If $Z$ has order $p$, then by fact (5), $Z$ must be cyclic. By Fact (4), $P/Z$ has order $p^2$. Further, by fact (3), $P/Z$ cannot be cyclic, because if it were cyclic, then $P$ would be abelian, which would mean that $Z=P$ has order $p^3$. Thus, $P/Z$ is a non-cyclic group of order $p^2$. By fact (6), it must be the elementary abelian group of order $p^2$.
CD5	If $Z$ has order $p^3$, $P$ is abelian.		$P$ has order $p^3$.		[SHOW MORE] We get $P=Z$, so $P$ is abelian.
CD6	We get the desired result.			Steps (CD3), (CD4), (CD5)	Step-combination.

Second part of proof: classifying the abelian groups

This classification follows from fact (7): the abelian groups of order $p^3$ correspond to partitions of 3, as indicated in the original statement of the classification.

Third part of proof: classifying the non-abelian groups

Given: A non-abelian group $P$ of order $p^3$. Let $Z$ be the center of $P$.

Previous steps: $Z$ is cyclic of order $p$, and $P/Z$ is elementary abelian of order $p^2$.

We first make some additional observations.

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
CN1	The derived subgroup (commutator subgroup) $P^{\prime }$ equals $Z$, so $P$ has class two.		$P$ is non-abelian of order $p^3$.	$P/Z$ is abelian, $Z$ has order $p$.	[SHOW MORE] Since $P/Z$ is abelian, $P^{\prime }$ is contained in $Z$. Since $Z$ has prime order, the only possible subgroups are the trivial subgroup and $Z$ itself. The derived subgroup cannot be trivial since $P$ is non-abelian, hence the derived subgroup must be $Z$. Note that this forces $P$ to have class two.
CN2	We can find elements $a,b\in P$ such that the images of $a,b$ in $P/Z$ are non-identity elements of $P/Z$ that generate it.			$P/Z$ is elementary abelian of order $p^2$	[SHOW MORE] $P/Z$ is elementary abelian of order $p^2$, hence is generated by a set of size two comprising non-identity elements. Choose $a,b$ to be arbitrary inverse images of these elements.
CN3	$Z,a,b$ together generate $P$.
CN4	$a$ and $b$ do not commute.			Steps (CN2), (CN3)	[SHOW MORE] If $a$ and $b$ commute, then $C_P(a)$ contains $Z,a,b$, hence equals all of $P$. Thus, $a\in Z$, so its image in $P/Z$ is the identity element, a contradiction to what we assumed.
CN5	Let $z=[a,b]$. Then, $z$ is a non-identity element of $Z$ and $⟨z⟩=Z$.			Steps (CN1), (CN4)	[SHOW MORE] We must have $z\in P^{\prime }$, which equals $Z$ by Step (1). By Step (4), $[a,b]$ is not the identity element. So $z$ is a non-identity element of $Z$.
CN6	The elements $a,b$ both have order either $p$ or $p^2$. Also, the elements $a^p$ and $b^p$ are both in $Z$.				[SHOW MORE] The order of $a$ must divide the order of the group, which is $p^3$. It cannot equal $p^3$, because that would make the group cyclic and hence abelian. It cannot equal $1$, because $a$ is a non-identity element. The only possibilities are $p$ and $p^2$. Also, the image of $a$ in $P/Z$ has order $p$ because $P/Z$ is elementary abelian, so $a^p\in Z$. The same goes for $b$.

We now make cases based on the orders of $a$ and $b$. Note that these cases may turn out to yield isomorphic groups, because the cases are made based on $a$ and $b$, and there is some freedom in selecting these.

Case A: $a$ and $b$ both have order $p$.

In this case, the relations so far give the presentation:

$⟨a,b,z\mid a^p=b^p=z^p=e,az=za,bz=zb,[a,b]=z⟩$

These relations already restrict us to order at most $p^3$, because we can use the commutation relations to express every element in the form $a^{\alpha }{b}^{\beta }{z}^{\gamma }$, where $\alpha ,\beta ,\gamma$ are integers mod $p$. To show that there is no further reduction, we note that there is a group of order $p^3$ satisfying all these relations, namely prime-cube order group:U(3,p). This is the multiplicative group of unipotent upper-triangular matrices with entries from the field of $p$ elements.

Thus, Case A gives a unique isomorphism class of groups. Note that the analysis so far works both for $p=2$ and for odd primes. The nature of the group obtained, though, is different for $p=2$, where we get dihedral group:D8 which has exponent $p^2$. For odd primes, we get a group of prime exponent.

Case B: $a$ has order $p^2$, $b$ has order $p$

In this case, we first note that $a^p\in Z=⟨z⟩$. Since $a^p$ is a non-identity element, there exists nonzero $r$ (taken mod $p$) such that $a^p=z^r$. Consider the element $c=b^r$ Then, by Fact (8), and the observation that $P$ has class two (Step (1) in the above table), we obtain:

$[a,c]=[a,b^r]=[a,b{]}^r=z^r=a^p$

Consider the presentation:

$⟨a,c\mid a^{p^2}=c^p=e,[a,c]=a^p⟩$

We see that all these relations are forced by the above, and further, that this presentation defines a group of order $p^3$, namely semidirect product of cyclic group of prime-square order and cyclic group of prime order.

Thus, there is a unique isomorphism class in Case B. Note that the analysis so far works both for $p=2$ and for odd primes. The nature of the group, though, is different for $p=2$, we get dihedral group:D8, which is the same isomorphism class as Case A.

Case B2: $a$ has order $p$, $b$ has order $p^2$.

Interchange the roles of $a,b$ and replace $z$ by $z^{-1}$ and we are back in Case B.

Case C: $a$ and $b$ both have order $p^2$.

By Fact (9), we can show that for odd prime, it is possible to make a substitution and get into Case B. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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For $p=2$, working out the presentation yields quaternion group.

Here is a summary of the cases:

Case letter	What it means	Isomorphism class of group for $p=2$	Isomorphism class of group for odd prime
A	Both $a,b$ have order $p$	dihedral group:D8	prime-cube order group:U(3,p)
B, B2	One of the elements has order $p^2$, the other has order $p$	dihedral group:D8	semidirect product of cyclic group of prime-square order and cyclic group of prime order
C	Both elements have order $p^2$	quaternion group	semidirect product of cyclic group of prime-square order and cyclic group of prime order

Finally, we note that:

• Dihedral group:D8 and quaternion group are non-isomorphic: The latter has no non-central element of order two, for instance.
• prime-cube order group:U(3,p) and semidirect product of cyclic group of prime-square order and cyclic group of prime order are non-isomorphic: The former has exponent $p$, the latter has exponent $p^2$.