Conditions for which Fischer \(F\)-subgroups in finite solvable groups are \(F\)-injectors (Q1969084)

Let \(\mathfrak F\) be a Fitting set of a solvable group \(G\). A Fischer \(\mathfrak F\)-subgroup of \(G\) belongs to \(\mathfrak F\) and contains every \(\mathfrak F\)-subgroup of \(G\) which it normalizes. An \(\mathfrak F\)-injector of \(G\) is a subgroup belonging to \(\mathfrak F\) that intersects every subnormal subgroup \(K\) of \(G\) in an \(\mathfrak F\)-maximal subgroup of \(K\). In a solvable group each \(\mathfrak F\)-injector is a Fischer \(\mathfrak F\)-subgroup. Dark showed that a Fisher \(\mathfrak F\)-subgroup need not be an \(\mathfrak F\)-injector. A subgroup \(A\) is locally pronormal in \(G\) if for each prime \(r\) dividing \(|A|\), a Sylow \(r\)-subgroup of \(A\) is pronormal in \(G\). A subgroup \(A\) is subnormally embedded in \(G\) if for each prime \(r\) dividing \(|A|\), a Sylow \(r\)-subgroup of \(A\) is a Sylow \(r\)-subgroup of some subnormal subgroup of \(G\). A subgroup \(A\) is normally embedded in \(G\) if for each prime \(r\) dividing \(|A|\), a Sylow \(r\)-subgroup of \(A\) is a Sylow \(r\)-subgroup of some normal subgroup of \(G\). In this paper it is shown that if a Fisher \(\mathfrak F\)-subgroup \(V\) is locally pronormal or subnormally embedded in \(G\) then \(V\) is an \(\mathfrak F\)-injector of \(G\). It is also shown that a subnormally embedded \(\mathfrak F\)-injector of \(G\) is normally embedded in \(G\).