a subgroup $x$ of a group $g$ is almost normal if the index $|g:n_g(x)|$ is finite, while $x$ is nearly normal if it has finite index in the normal closure $x^g$. this paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal.

In this note, we prove that the projective normality of (P(V )/G,L), the celebrated theorem of Erdös-Ginzburg-Ziv and normality of an affine semigroup are all equivalent, where V is a finite dimensional representation of a finite cyclic group G over C and L is the descent of the line bundle O(1)⊗|G|.