Exhaustion 2-subsets in dihedral groups of order 2p

Abstract

Let [Formula: see text] be the dihedral group of order [Formula: see text], where [Formula: see text] is an odd prime. A nonempty subset [Formula: see text] of [Formula: see text] is said to be exhaustive if there exists a positive integer [Formula: see text] such that [Formula: see text] covers all elements in [Formula: see text]. The smallest such [Formula: see text] is called the exhaustion number of [Formula: see text]. In general, finding [Formula: see text] for an arbitrary subset [Formula: see text] of [Formula: see text] is an interesting but difficult task. In this paper, we classify all possible exhaustion 2-subsets in [Formula: see text] by considering a 2-subset [Formula: see text] of [Formula: see text] with either [Formula: see text] or [Formula: see text] and [Formula: see text]. Some explicit formulas for [Formula: see text] are identified and hence some bounds are derived to prove the existence for certain families of exhaustive sets.