On the generalized $\sigma$-Fitting subgroup of finite groups

Let

\sigma =\{\sigma_{i} | i\in I\}

be some partition of the set

\mathbb P

of all primes, and let

G

be a finite group. A chief factor

H/K

of

G

is said to be

\sigma

-
central
(in

G

) if the semidirect product

(H/K) \rtimes (G/C_{G}(H/K))

is a

\sigma _{i}

-group for some

i=i(H/K)

; otherwise, it is called

\sigma

-
eccentric
(in

G

). We say that

G

is:

\sigma

-
nilpotent
if every chief factor of

G

is

\sigma

-central;

\sigma

-
quasinilpotent
if for every

\sigma

-eccentric chief factor

H/K

of

G

, every automorphism of

H/K

induced by an element of

G

is inner. The product of all normal

\sigma

-nilpotent (respectively

\sigma

-quasinilpotent) subgroups of

G

is said to be the

\sigma

-
Fitting subgroup
(respectively the
generalized

\sigma

-
Fitting subgroup
) of

G

and we denote it by

F_{\sigma}(G)

(respectively by

F^{*}_{\sigma}(G)

). Our main goal here is to study the relations between the subgroups

F_{\sigma}(G)

and

F^{*}_{\sigma}(G)

, and the influence of these two subgroups on the structure of

G

.