We define several graphs related to the p-blocks of a solvable group. We bound the diameter of these graphs when the defect group associated with the block is either abelian or normal and when the group has odd order. We give examples to show that these bounds are met.

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Suppose G is a p-solvable group, where p is odd. We explore the connection between lifts of Brauer characters of G and certain local objects in G, called vertex pairs. We show that if χ is a lift, then the vertex pairs of χ form a single conjugacy class. We use this to prove a sufficient condition for a given pair to be a vertex pair of a lift and to study the behaviour of lifts with respect to normal subgroups.

We count the number of lifts of an irreducible π-partial character that lies in a block with a cyclic defect group.

Let $H$ be a Hall $\pi$-subgroup of a finite $\pi$-separable group $G$, and let $\alpha$ be an irreducible Brauer character of $H$. If $\alpha(x)=\alpha(y)$ whenever $x,y \in H$ are $p$-regular and $G$-conjugate, then $\alpha$ extends to a Brauer character of $G$.

AMS 2000 Mathematics subject classification: Primary 20C15; 20C20

If G is a finite solvable group and p is a prime, then the normalizer of a Sylow p-subgroup has a normal Sylow 2-subgroup if and only if all non-trivial irreducible real 2-Brauer characters of G have degree divisible by p.

Suppose that $N$ is a normal $p$-subgroup of a finite group $G$ and let $G^0$ be the set of elements of $G$ whose $p$-part lies in $N$. We prove the existence of a canonical basis ${\rm IBr}(G, N)$ of the space of complex class functions of $G$ defined on $G^0$, such that the restriction $\chi^0$ of any irreducible complex character $\chi$ of $G$ is a linear combination $\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ of the elements of this basis, where the $d_{\chi \phi}$ are non-negative integers. Furthermore, if we write $\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$, then the $\Phi_\phi$ form the K\"ulshammer--Robinson ${\Bbb Z}$-basis of the ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules, where $R$ is a certain complete discrete valuation ring. By using these `decomposition numbers', it is possible to define a linking in the set of the irreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.

Suppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.