Problem Statement
Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.
Estimate $h(n)$ as well as possible.
Estimate $h(n)$ as well as possible.
Categories:
Group Theory
Progress
Pyber [Py87] has proved there exist constants $c_2>c_1>1$ such that $c_1^n<h(n)<c_2^n$. Erdős [Er97f] writes that the lower bound was already known to Isaacs.Source: erdosproblems.com/117 | Last verified: January 13, 2026