An Abelian cover of a group 
is a collection of subgroups
that are Abelian and whose union
is 
. Let 
be the least size of such a cover and let 
be the largest size of a set
of pairwise noncommuting group elements. For a non-Abelian
group, these are respectively the chromatic
number and clique number of its noncommuting
graph (Lecomte 2026).
Lecomte (2026) proved that
 |
for every integer 
, where 
is the floor function
and the supremum ranges over all groups. Writing 
, a group with 
attains equality precisely when 
and 
, where 
is the commutator subgroup
and 
is the group center. Here 
is the cyclic group of order
two and 
is the finite field with two elements.
Lecomte (2026) acknowledges ChatGPT and Aristotle assistance with exploratory proof
checking and consistency review.
