Products and joins of locally normal Fitting classes

Let $\pi$ be a nonempty set of primes. A nontrivial Fitting class $\mathfrak{F}$ is said to be normal in the class $\mathfrak{S}_\pi$ of all finite soluble $\pi$-groups or $\pi$-normal (we write $\mathfrak{F\trianglelefteq S}_\pi$) if $\mathfrak{F\subseteq S}_\pi$ and the $\mathfrak{F}$-radical of every $\pi$-group $G$ is a $\mathfrak{F}$-maximal subgroup of $G$. If $\pi$ is the set of all primes, then $\mathfrak{F}$ is called normal. The product $\mathfrak{FH}$ of Fitting classes $\mathfrak{F}$ and $\mathfrak{H}$ is called $\pi$-normal if $\mathfrak{FH}$ is a $\pi$-normal Fitting class. We prove the existence of $\pi$-normal products of Fitting classes factorizable by non-$\pi$-normal factors. Assume that $\mathbb{P}$ is the set of all primes, $\varnothing\neq\pi\subseteq\mathbb{P}$, $\mathfrak{F}$ is some Fitting class of $\pi$-groups, and $\omega=\sigma(\mathfrak{F})$ is the set of all prime divisors of all groups from $\mathfrak{F}$. It is proved that if $\mathfrak{F^2=F}$ and $\mathfrak{H}$ is the class of all $\pi$-groups with central $\omega$-socle, then the product $\mathfrak{FH}$ is $\pi$-normal although each of the factors $\mathfrak{F}$ and $\mathfrak{H}$ is not $\pi$-normal. The lattice join $\mathfrak{F\vee H}$ of Fitting classes $\mathfrak{F}$ and $\mathfrak{H}$ is the Fitting class generated by $\mathfrak{F\cup H}$. If $\mathfrak{F\vee H}$ is a $\pi$-normal Fitting class, then $\mathfrak{F\vee H}$ is called $\pi$-normal. Let $\mathfrak{F}$ and $\mathfrak{H}$ be Fitting classes of $\pi$-groups. We prove that the lattice join $\mathfrak{F\vee H}$ is a $\pi$-normal Fitting class if and only if $\mathfrak{F}$ or $\mathfrak{H}$ is a $\pi$-normal Fitting class.