p-groups with small number of character degrees and their normal subgroups

If G be a finite p-group and is a non-linear irreducible character of G, then (1) |G/Z (G) |^1{2}. In fernandez2001groups, Fern\'andez-Alcober and Moret\'o obtained the relation between the character degree set of a finite p-group G and its normal subgroups depending on whether |G/Z (G) | is a square or not. In this paper we investigate the finite p-group G where for any normal subgroup N of G with G' N either N Z (G) or |NZ (G) /Z (G) | p and obtain some alternate characterizations of such groups. We find that if G is a finite p-group with |G/Z (G) |=p^2n+1 and G satisfies the condition that for any normal subgroup N of G either G' N or N Z (G), then cd (G) =\1, p^{n\}. We also find that if G is a finite p-group with nilpotency class not equal to 3 and |G/Z (G) |=p^2n and G satisfies the condition that for any normal subgroup N of G either G' N or |NZ (G) /Z (G) | p, then cd (G) \1, p^{n-1, p^n\}.