Some Conditions for the Embeddability of Semigroups in Groups

We indicate two new classes of group embeddable semigroups, containing some earlier-known classes of semigroups with this property (Theorems 1 and 3), and construct examples of nonembeddable semigroups of class $K_{p,r}^2$ for $p+r\ge6$ (Theorem 2), thus disproving an earlier conjecture.