Groups and their coset monoids

This paper studies groups and their coset monoids. The anti-abnormal subgroups of a group are firstly introduced and investigated. It is shown that a group is an N-group if and only if each subgroup of it is anti-abnormal. Also, it is proved that the coset semigroups of N-groups are exactly the E-reflexive inverse semigroups which are factorisable and the natural connection between their semilattice of idempotents and lattice of subgroups of their group of units is a dual isomorphism. Finally, some characterizations of the coset semigroups of S-groups (U-groups and residually central groups respectively) are given. This extends McAlister's result in 1980.