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.
