Classification of Lie rings of prime-square order

Statement

Let $p$ be a prime number. There are only three possibilities, up to isomorphism, for a Lie ring of prime-square order $p^{2}$:

1. Abelian Lie ring whose additive group is cyclic group of prime-square order.
2. Abelian Lie ring whose additive group is elementary abelian group of prime-square order.
3. Nontrivial semidirect product of Lie rings of prime order.

Facts used

1. Structure theorem for finitely generated abelian groups

Proof

First part: identifying the possible additive groups

By Fact (1), the possibilities for the additive group, namely, the abelian groups of order $p^{2}$, correspond precisely to the possible partitions of the number $2$:

Partition	Corresponding abelian group
2	cyclic group of prime-square order $\mathbb {Z} _{p^{2}}$
1 + 1	elementary abelian group of prime-square order $\mathbb {Z} _{p}\times \mathbb {Z} _{p}$

Second part: dealing with the case of the cyclic group of prime-square order

If the additive group of a Lie ring is cyclic, then the Lie ring must be abelian, because the Lie bracket, being alternating, must vanish on any cyclic subgroup. Thus, for the case where the additive group of cyclic of prime-square order, the only possibility is that of an abelian Lie ring.

Third part: dealing with the case of the elementary abelian group of prime-square order

One possibility is that we have an abelian Lie ring. Let's consider the case that the Lie ring is non-abelian and try to deduce its structure.

Step no.	Assertion/construction	Previous steps used	Explanation
1	The additive group is a two-dimensional vector space over the field of $p$ elements		By definition of elementary abelian.
2	The derived subring is a one-dimensional vector space over the field of $p$ elements	Step (1)	Clearly, the derived subring must be nonzero for the Lie ring to be non-abelian. However, since the Lie bracket is alternating, it must be a quotient of the second exterior power of the original Lie ring. The second exterior power of a two-dimensional vector space has dimension ${\binom {2}{2}}=1$. Since the derived subring is nontrivial, it must have dimension exactly 1.
3	We can choose a basis $\{x,y\}$ for the Lie ring with one element generating the derived subring and the other element $y$ outside it.	Steps (1),(2)	Choose separately a nonzero element of the derived subring and an element outside the derived subring.
4	We must have $[x,y]=ax$ where $a$ is a nonzero element in the field of order $p$.	Steps (2), (3)
5	Pick $w=y/a$. Then, $\{x,w\}$ is a basis satisfying $[x,w]=x$.	Step (4)
6	The Lie ring we have gotten is unique and is isomorphic to the nontrivial semidirect product of Lie rings of prime order