Saturated Formations Closed Under Sylow Normalizers

Abstract

In this article we show that a finite soluble group possesses nilpotent Hall subgroups for well-defined sets of primes if and only if its Sylow normalizers satisfy the same property. In fact, this property of groups provides a characterization of the subgroup- closed saturated formations, whose elements are characterized by the Sylow normalizers belonging to the class, in the universe of all finite soluble groups.

Full text

Communications in Algebra
®
, 33: 2801–2808, 2005
Copyright © Taylor & Francis, Inc.
ISSN: 0092-7872 print/1532-4125 online
DOI: 10.1081/AGB-200065377
SATURATED FORMATIONS CLOSED UNDER
SYLOW NORMALIZERS
#
A. D’Aniello, C. De Vivo, and G. Giordano
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”,
Napoli, Italy
M. D. Pérez-Ramos
Departament d’Àlgebra, Universitat de València, Burjassot (València), Spain
In this article we show that a finite soluble group possesses nilpotent Hall subgroups
for well-defined sets of primes if and only if its Sylow normalizers satisfy the same
property. In fact, this property of groups provides a characterization of the subgroup-
closed saturated formations, whose elements are characterized by the Sylow normalizers
belonging to the class, in the universe of all finite soluble groups.
Key Words: Finite groups; Saturated formations; Sylow normalizers.
Mathematics Subject Classification: 20D10; 20D20.
1. INTRODUCTION
In the theory of finite groups, it is well-known that local properties, such as
those of p-subgroups and their normalizers, influence the global structure of the
groups. A vast classical theory rests on this fact. From this context, a starting point
of our study is the result of Glaubermann (1970), which states that a group is
a p-group, for some prime p, provided its Sylow subgroups are self-normalizing.
An extension of this result by Bianchi et al. (1986) says that a group is nilpotent
if its Sylow normalizers, i.e., the normalizers of Sylow subgroups, are nilpotent.
It is an easy observation that an analogous result for the class of supersoluble
groups does not hold; the symmetric group of degree 4 is such an example. The
structure of groups whose Sylow normalizers are supersoluble has been investigated
in Fedri and Serena (1988) and Bryce et al. (1991). We are interested in the
following question: which properties of the Sylow normalizers are inherited by the
whole group? This and other related questions have also been considered by other
authors from various viewpoints (see Ballester-Bolinches and Shemetkov, 1999;
Received April 15, 2004; Revised and Accepted November 3, 2004
#
Communicated by M. Dixon.
Address correspondence to M. D. Pérez-Ramos, Departament d’Àlgebra, Universitat de València,
Dr. Moliner 50, 46100 Burjassot (València), Spain; Fax: (34)96-3544735; E-mail: dolores.perez@uv.es
2801
Downloaded At: 20:09 9 January 2010

2802 D’ANIELLO ET AL.
D’Aniello et al., 2002, 2004). In this context, the class map n is defined as follows,
after some required notation. For a group G and a prime number p, a Sylow p-
subgroup of G will be denoted by G
p
. We denote by !"G# the set of all prime
numbers dividing the order of G. Then, for a class ! of groups,
n! $= "G $ N
G
"G
p
# ∈ !% for every prime p ∈ !"G##&
For notation and results about classes of groups and closure operations we
refer to Doerk and Hawkes (1992). In particular, if p is a prime, we denote by C
p
a
group of order p, and by GF(p) the finite field with p elements.
In the universe of all finite groups, it is easy to observe that if ! is an
homomorph (that is, q-closed), then n! is an homomorph, whereas nothing
analogous occurs for other frequently used closure operations. For instance, if ! is
the class of groups with an ordered Sylow tower, ! is an s-closed saturated Fitting
formation, but n! is closed under none of the operations r
0
, n
0
, s
n
: if p is a prime,
p ≥ 5, the direct product C
p
×Sym"4# does not belong to n!, while C
p
, Sym"4# ∈ n!;
also Alt"4# $ n! .
Moreover, if ! is s-closed, then clearly ! ⊆ n! , but the inclusion is strict
in general, as mentioned before, for the class of supersoluble groups. On the
other hand, the strict inclusion n! ⊂ ! may occur for classes !, which have very
few closure properties; for instance, the class of primitive groups (see D’Aniello
et al., 2002). Thus, it is not trivial to find classes ! which are n-closed, that is,
n! = ! .
In this article, we characterize the subgroup-closed saturated formations,
which are n-closed in the universe of finite soluble groups. These are exactly the
classes of groups with nilpotent Hall subgroups for suitable sets of primes. From
the point of view of saturated formations, they are locally defined by "
'"p#
, the class
of all soluble '"p#-groups for a set of primes '"p# containing p, for every prime p
in the characteristic of the formation, under a particular restriction on the sets of
primes '"p#. We call them covering-formations of soluble groups (see Definition). The
class of nilpotent groups and more generally, the classes of soluble groups, which are
direct product of Hall subgroups corresponding to pairwise disjoint sets of primes,
appear as very particular cases. (These classes of groups are called lattice formations
in Ballester-Bolinches et al., 2003.) As in our case, the saturated formations locally
defined by "
'"p#
, or by #
'"p#
, the class of all '"p#-groups, with p ∈ '"p#, under
other different restrictions on the sets of primes '"p#, provide several extensions of
nilpotent groups in the theory of classes of groups. They arise from a wide variety of
frameworks, namely lattices of subgroups, $ -critical groups, Frobenius p-nilpotence
criterion, factorized groups, formations and Fitting classes. Lattice formations are
a significant case of this fact. (See Ballester-Bolinches et al., 2003 for an account
of this development.) In the universe of all finite groups, a covering-formation
is not n-closed in general. An example is shown in Remark 1(c). We do not
know whether this result holds for “lattice formations” in the finite universe. (See
Remark 1(a).)
All the groups considered in the article are finite.
Downloaded At: 20:09 9 January 2010

SATURATED FORMATIONS 2803
2. DEFINITIONS AND PRELIMINARY RESULTS
Let p and q be different primes. If V
p
is an irreducible and faithful C
q
-module
over GF(p), the semidirect product (V
p
)C
q
corresponding to the action of C
q
on V
p
will be denoted by *"p% q#. We recall that *"p% q# is unique up to isomorphism.
Proposition 1. Let $ be an s-closed saturated formation such that
"G ∈ " $N
G
"G
p
# ∈ $ % for every prime p ∈ !"G## ⊆ $ %
and let p and q be two primes in Char"$ #. Then *"p% q# ∈ $ implies *"q% p# ∈ $ .
Proof. Assume that *$= *"p% q# ∈ $ . Let V
q
be an irreducible and faithful
*-module over GF(q) and let X denote the corresponding semidirect product (V
q
)*.
If X
p
∈ Syl
p
"X# and X
q
∈ Syl
q
"X#, then N
X
"X
p
# ' * ∈ $ and N
X
"X
q
# = X
q
∈ $ ,
because q ∈ Char"$ # and $ is a saturated formation. By the hypothesis, it follows
that X ∈ $ . Let P be a subgroup of * of order p. Since V
q
is a faithful *-module,
there exists an irreducible and faithful P-submodule of V
q
, say W . Then WP ≤ X
and so WP ∈ s$ = $ ; this means that *"q% p# ∈ $ and we are done.
The symmetry of the behaviour of two primes obtained in Proposition 1 suggested
the idea of the following saturated formations, which we call covering-formations.
Definition. Let ' be a set of primes and let % $'→ P"'#, P"'# the power set of ',
be a function which associates with each prime p of ' a subset '"p# of ', satisfying
the following two conditions:
(a) p ∈ '"p#, for every p ∈ ';
(b) q ∈ '"p# implies p ∈ '"q#.
The covering-formation #
%
is defined as the saturated formation LF"f
%
#,
locally defined by the formation function f
%
given by:
f
%
"p# = #
'"p#
if p ∈ '% f
%
"p# =∅ if p +∈ '%
where #
'"p#
is the class of all '"p#-groups.
In the universe " of soluble groups, the saturated formation #
%
∩ " will be
denoted by "
%
and called a covering-formation of soluble groups. We notice that it
is locally defined by the formation function g
%
given by:
g
%
"p# = "
'"p#
if p ∈ '% g
%
"p# =∅ if p +∈ '%
where "
'"p#
is the class of all soluble '"p#-groups.
Remark 1. (a) We observe that Char"#
%
# = ' and "'"p##
p∈'
is a covering of the
set '. If "'"p##
p∈'
is a partition of ', then the groups in #
%
are characterized by
being direct products of Hall '"p#-groups, for the primes p ∈ '. As mentioned in the
introduction, whether these formations are n-closed in the universe # of all finite
groups is an open question. In any case, they are not characterized by this property
as shown in (c).
Downloaded At: 20:09 9 January 2010

2804 D’ANIELLO ET AL.
(b) The condition '"p# ⊆ ' loses no generality. Assume that we associate with
each prime p of ' a set of primes '"p#, such that p ∈ '"p#. Consider the formation
functions f and f
1
given by:
f"p# = #
'"p#
if p ∈ '% f"p# =∅ if p +∈ '+
f
1
"p# = #
'"p#∩'
if p ∈ '% f
1
"p# =∅ if p +∈ '&
Then notice that LF"f# = LF"f
1
#. Moreover, ' =
!
p∈'
"'"p# ∩ '#.
(c) It is easy to observe that, in general, a covering-formation #
%
does not need
to be n-closed. For example, let ' = ,2% 3% 5- and let % be the function given by
p -→ '"p#, with '"2# = ', '"3# = ,2% 3-, '"5# = ,2% 5-. The alternating group Alt"5#
belongs to n#
%
\#
%
.
We observe also that " , the class of all soluble groups, is not n-closed in # as
any minimal insoluble group shows.
On the other hand, if $ is a covering-formation such that 2 $ Char"$ #, then it
is easily deduced that n$ = $ in the universe # from the Feit-Thompson Theorem
and the Theorem in Section 3.
For the rest of the exposition, the universe is " , the class of all soluble groups.
So, “group” will stand for “finite soluble group” and, for a class ! of groups,
n! = "G ∈ " $N
G
"G
p
# ∈ !% for every prime p ∈ !"G##&
& denotes the class of all nilpotent groups.
3. CLASSIFICATION OF s-CLOSED AND n-CLOSED SATURATED
FORMATIONS IN THE UNIVERSE OF SOLUBLE GROUPS
Proposition 2. Let "
%
= LF"g
%
# be the covering-formation of soluble groups of
characteristic ', defined by the function % $'−→ P"'# given by p -→ '"p#. Then:
"i# s"
%
= "
%
and n"
%
= "
%
+
"ii# "
%
= "G ∈ "
'
$ Hall
,p%q-
"G# ⊆ & % for every p% q ∈ !"G# such that q +∈ '"p##
= "G ∈ "
'
$ Hall
!
"G# ⊆ & , for every set of primes ! ⊆ !"G#
such that if p% q ∈ ! and p += q, then p +∈ '"q##&
Proof. (i) The s-closure of "
%
follows from the s-closure of g
%
"p# = "
'"p#
, for
every p ∈ ' (cf. Doerk and Hawkes, 1992, IV.3.14). In particular, we have "
%
⊆
n"
%
. Assume that "
%
+= n"
%
and let G ∈ n"
%
\"
%
of minimal order. Since "
%
is a saturated formation, then G is primitive and G/N ∈ "
%
, where N = Soc"G# =
O
p
"G# for some prime p ∈ '. In the next paragraph we prove that q ∈ '"p# for every
q ∈ !"G#. Given this G/N = G/C
G
"N# ∈ "
'"p#
= g
%
"p#, which implies G ∈ "
%
; this
is, then, a contradiction, and concludes the proof of (i).
Assume that q ∈ !"G#, q += p, and let us consider G
p
∈ Syl
p
"G# and G
q
∈
Syl
q
"G# such that T$= G
p
G
q
≤ G. Since G ∈ n"
%
and s"
%
= "
%
, it is clear that
Downloaded At: 20:09 9 January 2010

SATURATED FORMATIONS 2805
N
T
"G
p
# and N
T
"G
q
# belong to "
%
. If N
T
"G
p
# and N
T
"G
q
# are nilpotent, then T is
nilpotent (see Bianchi et al., 1986) and so we obtain the contradiction G
q
≤ C
G
"N# =
N . Assume that N
T
"G
p
# is not nilpotent. Then, since N
T
"G
p
# ∈ "
%
, there exists a
p-chief factor R/S of L$= N
T
"G
p
# such that G
p
≤ C
L
"R/S# += L and L/C
L
"R/S# ∈
g
%
"p# = "
'"p#
. This implies q ∈ '"p#. If N
T
"G
q
# is not nilpotent, we can argue
analogously and obtain p ∈ '"q#, from which q ∈ '"p# by the definition of %.
(ii) Let
' = "G ∈ "
'
$ Hall
,p%q-
"G# ⊆ & % for every p% q ∈ !"G# such that q $ '"p##&
We first prove the inclusion "
%
⊆ '. Let G ∈ "
%
and let T ∈ Hall
,p%q-
"G#,
with p% q ∈ !"G# such that q $ '"p#, and so p $ '"q#. Since s"
%
= "
%
, we have that
T ∈ "
%
. For any p-chief factor R/S of T we have that T/C
T
"R/S# ∈ g
%
"p# = "
'"p#
.
Since q $ '"p#, T
q
≤ C
T
"R/S# and consequently T
q
≤ O
p
/
%p
"T#. Then T
q
0 T.
Analogously, we obtain T
p
0 T. Hence, T is nilpotent and G ∈ '.
Secondly, we prove the inclusion ' ⊆ "
%
. Assume that this is not true and
let G ∈ '\"
%
of minimal order. Since ' is a homomorph and "
%
is a saturated
formation, G is a primitive group and G/N ∈ "
%
, where N = Soc"G# = O
p
"G# for
some prime p ∈ !"G#. Assume that q ∈ !"G# with q += p. If q $ '"p#, there exist
G
p
∈ Syl
p
"G# and G
q
∈ Syl
q
"G# such that G
p
G
q
is a nilpotent Hall ,p% q--subgroup
of G. There follows G
q
≤ C
G
"N# = N ≤ G
p
and so the contradiction G
q
= 1. This
means that q ∈ '"p# and so G/N = G/C
G
"N# ∈ "
'"p#
= g
%
"p#, which implies the
contradiction G ∈ "
%
.
The remainder of the statement is easily proven.
The following theorem characterizes the s-closed, saturated formations that
are n-closed, in the universe of soluble groups.
Theorem. Let $ be an s-closed saturated formation of soluble groups. Then $ is
n-closed if and only if $ is a covering-formation of soluble groups.
Proof. The sufficient condition follows from (i) of Proposition 2.
Let $ be a saturated formation such that s$ = $ and n$ = $ . Let
Char"$ # = ' and let F be the canonical local definition of $ . For every p ∈ ',
let '"p# = Char($ "p#). Since the formation function F is integrated and full, we
have p ∈ '"p# ⊆ ', for every p ∈ '. Moreover, if q ∈ '"p#, i.e., C
q
∈ F"p#, then
*"p% q# ∈ LF"F# = $ ; consequently *"q% p# ∈ $ , by Proposition 1, and so p ∈ '"q# =
Char"F"q##. Thus the function % $'→ P"'#, given by p -→ '"p# = Char"F"p##,
satisfies the conditions (a) and (b) of the Definition of covering formations.
The statement will be proved if we show that $ = "
%
.
We prove next that "
'"p#
∩ $ = F"p#, for every p ∈ '.
This will conclude the proof, since then "
p
"g
%
"p# ∩ $ # = "
p
""
'"p#
∩ $ # =
"
p
F"p# = F"p# for every p ∈ ', which implies $ = LF"g
%
# = "
%
(cf. Doerk and
Hawkes, 1992, IV. 3.8(b)).
Since s$ = $ we know that sF"p# = F"p#, for every p ∈ ' (cf. Doerk and
Hawkes, 1992, IV.3.16) and so F"p# ⊆ "
'"p#
∩ $ , for every p ∈ '.
To prove the reverse inclusion, we will assume that F"p# += "
'"p#
∩ $ for some
prime p ∈ ' and produce a contradiction. This will be done in a number of steps.
Downloaded At: 20:09 9 January 2010

2806 D’ANIELLO ET AL.
The argument begins by choosing a group G ∈ ""
'"p#
∩ $ #\F"p# that is minimal in
the following sense.
Let ! = "G ∈ ""
'"p#
∩ $ #\F"p# $ 1"!"G# ∪ ,p-#\,p-1 is minimal#. Let us
consider now G ∈ ! such that 1G1 is minimal. By the choice of G, there exists a
unique minimal normal subgroup N of G and G/N ∈ F"p#. If N ≤ O
p
"G#, then
G ∈ "
p
F"p# = F"p#, a contradiction. Consequently N ≤ O
q
"G#, for some prime
q += p.
Step 1. We prove that p does not divide 1G1. Assume that this is not true
and let V
p
be a faithful G-module over GF"p#. Let X = (V
p
)G the corresponding
semidirect product. We see next that X belongs to $ .
Let r ∈ !"G#, r += p, and let G
r
∈ Syl
r
"G#. Then we have L$= N
X
"G
r
# =
C
V
p
"G
r
#N
G
"G
r
#. Since G ∈ $ = s$ , it is clear that N
G
"G
r
# ∈ $ . Assume now that
R/S is a p-chief factor of L such that R ≤ C
V
p
"G
r
#. Obviously C
V
p
"G
r
#G
r
≤ C
L
"R/S#
and so
L/C
L
"R/S# ' N
G
"G
r
#/C
N
G
"G
r
#
"R/S# ∈ "
r
/
∩ qs"G#&
By the choice of G we deduce that L/C
L
"R/S# ∈ F"p# and obtain L = N
X
"G
r
# ∈ $ .
On the other hand, we have N
X
"V
p
G
p
# = V
p
N
G
"G
p
#, G
p
∈ Syl
p
"G#. Since
O
q
/
"G# = 1 and p ∈ !"G# it is clear that N
G
"G
p
# += G. By the choice of G, there
follows that N
G
"G
p
# ∈ F"p# and so N
X
"V
p
G
p
# ∈ "
p
F"p# = F"p# ⊆ $ . Since n$ = $ ,
we obtain that X ∈ $ . Now O
p
/
"X# = 1 and so X ∈ F"p#, from which we get the
contradiction G ∈ F"p#. Thus, we have proved that p $ !"G#.
Step 2. Now the group G satisfies N ≤ F"G# = O
q
"G#. Since ."G# < F"G#,
there is a proper subgroup M of G such that G = O
q
"G#M. For such an M, let
( = ,T 0G$T ≤ O
q
"G# and G = TM- and let T ∈ ( be of minimal order.
The choice of T implies T
/
M += G. Let us consider now G3
reg
C
p
= G
/
C
p
, the
regular wreath product of G with C
p
, where G
/
= G ×···×G is the base group.
If we denote D"M# = ,"m% & & & % m# $ m ∈ M- and T
/
= T ×···×T ≤ G
/
, we
notice that (D"M#% C
p
) = 1 and (T
/
%C
p
) is normalized by D"M#C
p
. Moreover, we
have by Doerk and Hawkes (1992, A.18.4)
C
(T
/
%C
p
)
"C
p
# = C
T
/
"C
p
# ∩ (T
/
%C
p
) = ,"t% & & & % t# $ t ∈ T% t
p
∈ T
/
-
= ,"t% & & & % t# $ t ∈ T
/
-
= D"T
/
#
taking into account that p += q and T is a q-group.
Step 3. Consider now Y$= (T
/
%C
p
)D"M#C
p
.
We now prove that Y ∈ $ = "
'
∩ "
"
r∈'
"
r
/
F"r##.
Clearly Y ∈ "
'
. Let r ∈ '. If r $ !"G# ∪ ,p- then, obviously, Y ∈ "
r
/
⊆ "
r
/
F"r#.
Assume that r ∈ !"G# ∪ ,p-, but r += q. Then, either r = p or r ∈ !"G# ⊆ '"p#.
In the second case we have that p ∈ '"r# = Char"F"r##. So, in any case, C
p
∈ F"r#. Since
G ∈ $ = s$ , we have M ∈ $ . In particular, M ∈ "
r
/
F"r# and, therefore, D"M#C
p
=
D"M# × C
p
∈ "
r
/
F"r#. Moreover T ∈ "
q
so Y = (T
/
%C
p
)D"M#C
p
∈ "
r
/
F"r# as q += r.
Consider now the case r = q. Since G ∈ $ and O
q
/
"G# = 1, it is clear
that G ∈ F"q# = sF"q#, which implies M ∈ F"q#. Again q ∈ !"G# ⊆ '"p# implies
Downloaded At: 20:09 9 January 2010

SATURATED FORMATIONS 2807
p ∈ '"q# = Char"F"q##; and that is C
p
∈ F"q#. Then D"M#C
p
∈ F"q# and so
Y = (T
/
%C
p
)D"M#C
p
∈ "
q
F"q# = F"q# ⊆ "
q
/
F"q#.
Thus, Y ∈ $ is proven.
Step 4. Now let V
p
be a faithful Y -module over GF"p# and let us consider
Z$= (V
p
)Y the corresponding semidirect product. We claim that Z belongs to n$ = $ .
Since D"M#C
p
normalizes C
p
and C
(T
/
%C
p
)
"C
p
# = D"T
/
#, we notice that
N
Y
"C
p
# = N
(T
/
%C
p
)
"C
p
#D"M#C
p
= C
(T
/
%C
p
)
"C
p
#D"M#C
p
= "D"T
/
#D"M## × C
p
&
Now, D"T
/
#D"M# ' T
/
M += G, by the choice of T . On the other hand, the choice of
G implies that D"T
/
#D"M# ∈ F"p# and, clearly, N
Y
"C
p
# = "D"T
/
#D"M## × C
p
∈ F"p#.
Therefore, we have
N
Z
"V
p
C
p
# = V
p
N
Y
"C
p
# ∈ "
p
F"p#
= F"p# ⊆ $ &
Now let s ∈ !"Z#% s += p, and let Y
s
∈ Syl
s
"Y#. We have L$= N
Z
"Y
s
# = C
V
p
"Y
s
#N
Y
"Y
s
#.
Since Y ∈ $ =s$ then N
Y
"Y
s
# ∈ $ . Assume that R/S is a p-chief factor of L such
that R ≤ C
V
p
"Y
s
#. Obviously Y
s
C
V
p
"Y
s
# ≤ C
L
"R/S# and so
L/C
L
"R/S# ' N
Y
"Y
s
#/C
N
Y
"Y
s
#
"R/S# ∈ "
s
/
∩ qs"Y# ⊆ "
s
/
∩ "
'"p#
∩ $ &
Thus "!"L/C
L
"R/S## ∪ ,p-#\,p- is properly contained in !"G# and so the choice
of G implies that L/C
L
"R/S# ∈ F"p#. Then L = N
Z
"Y
s
# ∈ $ . Thus Z ∈ n$ = $ as
claimed.
Step 5. Since O
p
/
"Z# = 1, we have that Z ∈ F"p# = sF"p#. Consequently
(T
/
%C
p
)D"M# ∈ F"p#. We notice now that
,"t% t
−1
% 1%&&&%1#$t∈ T- ⊆ (T
/
%C
p
) =
#
"t
1
%&&&%t
p
#$t
i
∈ T%
p
$
i=1
t
i
∈ T
/
%
and let us consider 0$G
/
→ G, the canonical projection of G
/
on the first
component. It is clear that 0"(T
/
%C
p
)D"M## = G. Consequently,
G ∈ q"(T
/
%C
p
)D"M## ⊆ qF"p# = F"p#%
which provides the final contradiction.
Remark 2. In the Theorem, we cannot remove the hypothesis that $ is s-closed.
In fact, there exist n-closed saturated formations of soluble groups, which are not
covering-formations as the following example shows.
Let p and q be primes such that q divides p − 1, and let )
,p%q-
be the class
of supersoluble ,p% q--groups. The class $ $= n)
,p%q-
is a saturated formation
Downloaded At: 20:09 9 January 2010

2808 D’ANIELLO ET AL.
(cf. D’Aniello et al., 2004 or Fedri and Serena, 1988) and $ is n-closed because
)
,p%q-
is s-closed. Clearly, $ is not a covering-formation. It is easy to verify that $
is not s-closed.
ACKNOWLEDGMENT
The fourth author has been supported by Proyecto BFM2004-06067-C02-02,
Ministerio de Educacíon y Ciencia and FEDER, Spain.
REFERENCES
Glaubermann, G. (1970). Prime-power factor groups of finite groups II. Math. Z. 117:46–56.
Bianchi, M. G., Gillio Berta Mauri, A., Hauck, P. (1986). On finite groups with nilpotent
Sylow normalizers. Arch. Math. 47:193–197.
Fedri, V., Serena, L. (1988). Finite soluble groups with supersoluble Sylow normalizers.
Arch. Math. 50:11–18.
Bryce, R. A., Fedri, V., Serena, L. (1991). Bounds on the Fitting length of finite soluble
groups with supersoluble Sylow normalizers. Bull. Austral. Math. Soc. 44:19–31.
Ballester-Bolinches, A., Shemetkov, L. A. (1999). On normalizers of Sylow subgroups in
finite groups. Siberian Math. J. 40(1):1–2.
D’Aniello, A., De Vivo, C., Giordano, G. (2002). Finite groups with primitive Sylow
normalizers. Boll. U.M.I. 87:3–13.
D’Aniello, A., De Vivo, C., Giordano, G. (2004). Saturated formations and Sylow
normalizers. Bull. Austral. Math. Soc. 69:25–33.
Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin-New York: Walter De Gruyter.
Ballester-Bolinches, A., Martínez-Pastor, A., Pedraza-Aguilera, M. C., Pérez-Ramos, M. D.
(2003). On nilpotent-like fitting formations. In: Proceedings of Groups-St Andrews 2001
in Oxford. Lecture Notes Series. London Mathematical Society. Vol. 1. Cambridge:
Cambridge University Press, pp. 31–38.
Downloaded At: 20:09 9 January 2010