Let $G$ be a finite group and $K$ be a field. Consider the group algebra $K[G]$, and for $k\in \{2,3\}$ look for a largest linear subspace $X\subset K[G]$ for which $X^k=0$ (in the sense that $x_1x_2\ldots x_k=0$ for all $x_1,x_2,\ldots,x_k$ in $X$).

Theorem-2. For $k=2$, ${\rm codim}\, X\geqslant |G|/2$.

Proof. The bilinear form $(x,y)\to [e]xy$ (where $x,y\in K[G]$ and $[e]x$ means the coefficient of $e$ in the expansion of $x\in K[G]$ as a linear combination of elements of $G$) is non-degenerated, thus, if two subspaces (in our situation both subspaces are $X$) are orthogonal, the sum of their codimensions is not less than $|G|$.

Theorem-3. For $k=3$, it may appear that ${\rm codim}\, X<|G|^{0.99}$ for arbitrarily large $|G|$.

Proof. Say, for $G=C_2^n$, where $C_2$ is the cyclic group of order 2, denote the generator of $i$-th $C_2$ by $g_i$, and take for $X$ the span of elements $\prod_{i\in I} (g_i-e)$, where $I\subset \{1,\ldots,n\}$ and $|I|>n/3$. Since $(g_i-e)^2=0$ for all $i$, by pigeonhole principle we get $X^3=0$, and the codimension of $X$ is $\sum_{k\leqslant n/3} {n\choose k}<2^{0.92 n}$ by the entropy bound for binomial distribution.

This example may seem artificial, but actually the minimal codimension of the subspace with zero cube is an upper bound for many natural combinatorial characteristics of the group, like (the half of) the maximal size of a set without progressions of length 3, it is a group algebra view of the breakthrough paper by Croot--Lev--Pach.