A 2-covering for a finite group G is a set of proper subgroups of G such that every pair of elements of G is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group G is called the 2-covering number and denoted by (Formula presented.) In [3] it is conjectured that if G is solvable and not 2-generated, then (Formula presented.) where q is a prime power. We disprove this conjecture.
2-covering numbers of some finite solvable groups
Lucchini A.
2026
Abstract
A 2-covering for a finite group G is a set of proper subgroups of G such that every pair of elements of G is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group G is called the 2-covering number and denoted by (Formula presented.) In [3] it is conjectured that if G is solvable and not 2-generated, then (Formula presented.) where q is a prime power. We disprove this conjecture.
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