# Chief factors and local formation

Every thing below is concerned with finite groups.

A class of groups is a collection $\mathcal{X}$ of groups with the property that if $G \in \mathcal{X}$ and if $H \cong G$, then $H \in \mathcal{X}$.

A class of groups $\mathcal{F}$ is called a formation, provided that the following conditions are satisfied:

(1) If $G \in \mathcal{F}$, then $G/N \in \mathcal{F}$ for any normal subgroup $N$ of $G$,

(2) If $G/M$ and $G/N$ are both in $\mathcal{F}$, then $G/(M \cap N) \in \mathcal{F}$ for any normal subgroups $M$ and $N$ of $G$.

Now let $\Pi$ be the set of all prime numbers. Then, by a formation function $f$, we mean a function $f$ define on $\Pi$ such that $f(P)$, possibly empty, is a formation. A chief factor $H/K$ of a group $G$ is called $f$-central in $G$ if $G/C_{G}(H/K) \in f(p)$ for all primes $p$ dividing $|H/K|$. A formation is a local formation if there exists a formation function $f$ such that $\mathcal{F}$ is a local formation defined by $f$ and we write $\mathcal{F}=LF(f)$ and call $f$ a local definition of $\mathcal{F}$.

Among all the possible local definitions for a local formation , there exists exactly one of them, denoted it by $F$, such that $F$ is integrated (i.e., $F(p) \subseteq \mathcal{F}$ for all $p \in \Pi$) and full (i.e., $\mathcal{S}_{p}F(p) = F(p)$ for all $p \in \Pi$) where $\mathcal{S}_{p}$ is the class of $p$-groups.

How did the author used Lemma 2.3 (see the lemma)

in this portion of the proof of Corollary 3.2. (see the portion)?

My question is exactly in the line that is underlined with blue. How did the authors conclude that $G/C_{G}(K_{1}/K_{2}) \in F_{1}(q)$?

I know that one possible local definition for the class of $p$-nilpotent groups, denoted by $\mathcal {N}_{p}$, is the following: We define the formation function as follows: $f(p)=(1)$ and $f(q)=\mathcal {G}$ for $q \neq p$, Where $(1)$ is the class that contains only the trivial group $\{1\}$ and $\mathcal{G}$ is the class of all finite groups.

According to the above local definition of $p$-nilpotent groups $G/C_{G}(K_{1}/K_{2}) \in f(q)$ clearly as $f(q)= \mathcal{G}$ the class of all finite groups. But the problem is that is this local definition is not integrated as $\mathcal{G}$ is not contained in the class of $p$-nilpotent groups $\mathcal {N}_{p}$.

I think that the full and integrated formation function (which is unique) $F_{1}$ for the class of $p$-nilpotent $\mathcal {N}_{p}$ groups is given by $F_{1}(p)=\mathcal{S}_{p}$ and $F_{1}(q)=\mathcal {N}_{p}$ for $q \neq p$, Where ${S}_{p}$ is the class of all finite $p$-groups and $\mathcal {N}_{p}$ is the class of all $p$-nilpotent groups. However I do not know how to prove that $G/C_{G}(K_{1}/K_{2})$ is $p$-nilpotent so that $G/C_{G}(K_{1}/K_{2}) \in \mathcal {N}_{p} = F_{1}(q)$.