𝑝-groups with exactly four codegrees

Let G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)}. Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree {p^{2}}, {\lvert G:G^{\prime}\rvert=p^{2}}, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by {p^{7}}. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by {p^{10}}.

A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎.

On the 𝑝-closedness and 𝑝-nilpotency of finite groups

Let {\operatorname{acd}(G)} and {\operatorname{acs}(G)} denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if \operatorname{acd}(G)<2(p+1)/(p+3) , then G=O_{p}(G)\times O_{{p^{\prime}}}(G) , and that if \operatorname{acs}(G)<4p/(p\kern-1.0pt+\kern-1.0pt3) , then G=O_{p}(G)\kern-1.0pt\times\kern-1.0ptO_{{p^{\prime}}}(G) with {O_{p}(G)} abelian, where p is a prime.

The solvability comes from a given set of character degrees

Let [Formula: see text] be a finite group and the irreducible character degree set of [Formula: see text] is contained in [Formula: see text], where [Formula: see text], and [Formula: see text] are distinct integers. We show that one of the following statements holds: [Formula: see text] is solvable; [Formula: see text]; or [Formula: see text] for some prime power [Formula: see text].

The largest character degree and the Sylow subgroups of finite groups

Let [Formula: see text] be a Sylow [Formula: see text]-subgroup and [Formula: see text] be the largest irreducible character degree of a finite nonabelian group [Formula: see text]. Then [Formula: see text].

On the Largest Irreducible Character Degree of a Finite Solvable Group

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn.