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

Proof

First part of proof: showing that either the group is abelian or it is of class two with a particular center and inner automorphism group

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
1	$Z$ is nontrivial	Fact (1)	$P$ has order $p^{3}$, specifically, a power of a prime		Fact+Given direct
2	The order of $Z$ cannot be $p^{2}$	Facts (2), (3), (4)	$P$ 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 $p^{3}$.
3	The order of $Z$ is either $p$ or $p^{3}$	Fact (4)	$P$ has order $p^{3}$	Steps (1), (2)	By fact (4), the order of $Z$ must divide the order of $P$. The only possibilities are $1,p,p^{2},p^{3}$. Step (1) eliminates the possibility of $1$, and step (3) eliminates the possibility of $p^{2}$. This leaves only $p$ or $p^{3}$.</toggledisplay>
4	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 $p^{2}$.
5	If $Z$ has order $p^{3}$, $P$ is abelian.		$P$ has order $p^{3}$.		[SHOW MORE] We get $P=Z$, so $P$ is abelian.
6	We get the desired result.			Steps (3), (4), (5)	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

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