Lifts and vertex pairs in solvable groups

Abstract

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.

Anchors of irreducible characters

“…Let (W, γ) be a nucleus of χ such that Q is a Sylow p-subgroup of W and δ = Res W Q (α), where γ = αβ, with α a p ′ -special character and β a p-special character of W (cf. [17,Sections 2,3]). By Lemma 6.3, the trivial Brauer character is a constituent of β • .…”

“…In [17] Navarro associated, via the theory of special characters, to each ordinary irreducible character χ of a p-solvable group G, a G-conjugacy class of pairs (Q, δ), where Q is a p-subgroup of G and δ is an ordinary irreducible character of Q, which behave in certain ways as the Green vertices of indecomposable modules (see also [5], [2] and [3]). We call such a pair (Q, χ) a Navarro vertex of χ.…”

“…Let p = 2, G = GL(2,3) and N = SL(2,3). Let R be the unique Sylow 2-subgroup of N and H a complement of R in N .…”

Anchors of irreducible characters

“…Let (W, γ) be a nucleus of χ such that Q is a Sylow p-subgroup of W and δ = Res W Q (α), where γ = αβ, with α a p ′ -special character and β a p-special character of W (cf. [17,Sections 2,3]). By Lemma 6.3, the trivial Brauer character is a constituent of β • By Lemma 6.2, there exists an OW -lattice X affording β and with vertex Q.…”

Navarro vertices and lifts in solvable groups

“…For each irreducible complex character ∈ Irr( ), Navarro [17] introduced a canonical pair ( , ) associated to which is uniquely defined up to -conjugacy, where is a -subgroup of and ∈ Irr( ). Such a pair ( , ) is called a Navarro vertex of , and has been shown to be quite useful in many problems in the character theory of solvable groups (see, for example, [2,3,7,8,9,14,18,19] and the references therein).…”