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

If g be a finite p-group and χ is a non-linear irreducible character of g, then χ(1) ≤ |g/z(g)| 12 . in [2], 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)| = p2n+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, pn}. we also find that if g is a finite p-group with nilpotency class not equal to 3 and |g/z(g)| = p2n 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, pn−1, pn}.