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Let $$\pi $$ be a set of prime numbers and G be a $$\pi $$ -separable group. If $$\varphi (1)_{\pi '}^{2}$$ divides $$|G: \ker \varphi |_{\pi '}$$ for every $$\pi $$ -partial character $$\varphi \in \mathrm{I}_{\pi }(G)$$ , then G has a normal Hall $$\pi '$$ -subgroup, where $$\varphi (1)_{\pi '}$$ denotes the $$\pi '$$ -part of $$\varphi (1)$$ and $$\mathrm{I}_{\pi }(G)$$ is the set of irreducible $$\pi $$ -partial characters of G.

Let G be a finite group, p be a prime and $$\mathrm{Irr}(G)$$ be the set of irreducible (complex) characters of G. Let $$\chi \in \mathrm{Irr}(G)$$ and write $$\mathrm{cod}(\chi )=|G: \ker \chi |/\chi (1)$$ , where $$\ker \chi $$ is the kernel of $$\chi $$ . We prove in this note that if G is solvable and $$\mathrm{cod}(\chi )$$ is a $$p'$$ -number for every monomial character $$\chi \in \mathrm{Irr}(G)$$ , then G has a normal p-complement.

We show that the simple unitary groups $$U_3(q)$$ and $$U_4(q)$$ , with $$2< q < 10^2$$ a prime power, are characterized among all groups by their order and the degrees of the vertices of their prime graph.

Let G be a finite group and p be a prime. We prove in this note that if every irreducible monolithic p-Brauer character of G is monomial then G is solvable.Communicated by J. Zhang

Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$-subgroup of $G$. We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Let G be a finite group and p be a fixed prime. A p-Brauer character of G is said to be monomial if it is induced from a linear p-Brauer character of some subgroup (not necessarily proper) of G. Denote by IBrm(G) the set of irreducible monomial p-Brauer characters of G. Let H = G′Op′ (G) be the smallest normal subgroup such that G/H is an abelian p′-group. Suppose that g ∈ G is a p-regular element and the order of gH in the factor group G/H does not divide |IBrm(G)|. Then there exists φ ∈ IBrm(G) such that φ(g) = 0.

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G / P {G/P} is nilpotent if and only if φ ⁢ ( 1 ) 2 {\varphi(1)^{2}} divides | G : ker ( φ ) | {|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G.

Let G be a finite group and p a prime divisor of | G |. Denote by IBr(G) and LBr(G) the sets of irreducible p-Brauer characters and linear p-Brauer characters of G, respectively. It is known that IBr(G) = LBr(G) if and only if P ∈ Sylp(G) is normal and G/P is abelian. We consider the local situation in this note. Let B be a p-block of G. We characterize the finite groups G with IBr(B) ⊆ LBr(G). And we get some results related to those of Holm, Willems and Isaacs, Smith.

We determine the groups with exactly two nonlinear non-faithful irreducible characters whose kernels intersect trivially.

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$-Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

Let p be a given prime. A finite group G whose all nonlinear irreducible p-Brauer characters are real valued is called a p-regular ℝ1-group. The aim of this article is to show some results about the structures of p-regular ℝ1-groups. In particular, we will build the connections between p-regular ℝ1-groups and p-modular Frobenius groups. If these results apply to a finite group G with p∤|G|, we obtain similar results about finite groups whose nonlinear irreducible characters are real valued, and establish connections between these groups and Frobenius groups.