# Existence of abelian ideals of small prime power order in nilpotent Lie ring

## Statement

Suppose $p$ is a prime number and $G$ is a nilpotent Lie ring of order $p^n$. Then, if $k$ is a nonnegative integer such that $n \ge 1 + k(k-1)/2$ (i.e., $n > k(k-1)/2$), $G$ has an Abelian ideal (?) of order $p^k$.

## Proof

Outline: We use Fact (1) to show that there is an abelian ideal of order at least $p^k$, and then use Fact (2) to find within that an abelian ideal of order exactly $p^k$.