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}.