Relative normal complements in finite groups

Abstract:

$(G,H,{H_0},\pi )$ denotes the following configuration: $H$ and ${H_0}$ are the subgroups of the finite group $G$ with ${H_0} \trianglelefteq H$ is the set of primes dividing $(H:{H_0})$. For $(G,H,{H_0},\pi )$ we consider conditions $({\text {A}})$, $({{\text {B}}_0})$, and $({\text {C}})$: $({\text {A}})$ Any two $\pi$-elements of $H - {H_0}$ which are $G$-conjugate are $H$-conjugate. $({{\text {B}}_0})$ For each $\pi$-element $x \in H - {H_0}$, ${C_G}(x) = I(x){C_H}(x)$ where $I(x)$ is a normal $\pi ’$-subgroup of ${C_G}(x)$. $({\text {C}})\left | {{{(H - {H_0})}^{G,\pi }}} \right | = (G:H)\left | {H - {H_0}} \right |$. We show that if $(G,H,{H_0},\pi )$ satisfies $({{\text {B}}_0})$ and $({\text {C}})$, or $({\text {A}})$ and $({{\text {B}}_0})$, and if $H/{H_0}$ is solvable, then there is a unique relative normal complement ${G_0}$ of $H$ over ${H_0}$.