Inverse Glauberman-Isaacs correspondence and subnormal subgroups. (Q254334)

Let \(A\) be a finite group acting coprimely on a finite group \(G\). One way to view the well-known Glauberman-Isaacs correspondence is the following: Let \(B_1\trianglelefteq B_2\leq A\). Then there exists a bijection from the set of all \(B_2\)-invariant characters of \(C_G(B_1)\) onto the set \(\mathrm{Irr}(C_G(B_2))\). The author now considers the inverse map and studies what happens when the hypothesis \(B_1\trianglelefteq B_2\) is weakened to \(B_1\) just being subnormal in \(B_2\). It turns out that in this situation one can still define an injection from \(\mathrm{Irr}(C_G(B_2))\) into \(\mathrm{Irr}(C_G(B_1))\) which retains most of the good properties of the inverse of the Glauberman-Isaacs correspondence (and agrees with it in case that \(B_1\trianglelefteq B_2\)). The author remarks that the results of this paper will be used in a future paper to describe and control Clifford theory as it relates to \(p\)-local subgroups.