## Finite p-nilpotent groups with some subgroups c-supplemented

*J. Aust. Math. Soc. ***78 **(2005),

**FINITE ***p***-NILPOTENT GROUPS WITH SOME SUBGROUPS**
**c****-SUPPLEMENTED**
**XIUYUN GUO and K. P. SHUM**
(Received 10 June 2001; revised 18 February 2004)
A subgroup

*H *of a finite group

*G *is said to be

*c*-supplemented in

*G *if there exists a subgroup

*K *of

*G*such that

*G *=

*H K *and

*H *∩

*K *is contained in core

*G *.

*H */. In this paper some results for finite

*p*-nilpotentgroups are given based on some subgroups of

*P c*-supplemented in

*G*, where

*p *is a prime factor of theorder of

*G *and

*P *is a Sylow

*p*-subgroup of

*G*. We also give some applications of these results.

2000

*Mathematics subject classification*: primary 20D10, 20D20.

**1. Introduction**
Let

*G *be a finite group. The relationship between the properties of the Sylow sub-groups of

*G *and the structure of

*G *has been investigated by a number of authors(see, for example, In particular, Buckley in 1970 provedthat a group of odd order is supersolvable if all its minimal subgroups are normal.

Srinivassan proved that a finite group is supersolvable if every maximal subgroupof every Sylow subgroup is normal. These two important results on supersolvablegroups have been generalized by many authors. One direction of generalization is toreplace the normality condition of maximal subgroups or minimal subgroups of Sylowsubgroups by a weaker condition; and the other direction of generalization is to min-imize the number of maximal subgroups or minimal subgroups of Sylow subgroups.

The research of the first author was partially supported by a grant of the Shanghai Natural ScienceFoundation (Grant No. 03ZR) and a grant of the National Natural Science Foundation of China (GrantNo. 10171086). The research of the second author is partially supported by a UGC(HK) grant #2160126(2003/2005).

c 2005 Australian Mathematical Society 1446-8107/05 $A2:00 + 0:00
It has been observed that the property of ‘normality’ for some maximal subgroups orsome minimal subgroups of Sylow subgroups gave a lot of useful information on thestructure of groups. In this paper, we shall continue to study the structure of finitegroups on the assumption that some subgroups are

*c*-supplemented and obtain someinteresting results for finite

*p*-nilpotent groups. As an application of our results, wegive conditions for a finite group to be in a saturated formation containing the class offinite supersolvable groups.

Throughout this paper, all groups are finite. Our terminology and notation are

**2. Preliminaries**
A subgroup

*H *of a group

*G *is said to be

*c*-supplemented in

*G *if there exists a
subgroup

*K *of

*G *such that

*G *=

*H K *and

*H *∩

*K *≤ core .

several lemmas for later use in this paper.

LEMMA 2.1 Lemma 2.1]).

*Let H be a subgroup of a group G. Then the*
(1)

*Let K be a subgroup of G such that H is contained in K . If H is c-supplemented*
*in G then H is c-supplemented in K .*
(2)

*Let N be a normal subgroup of G such that N is contained in H . Then H is*
*c-supplemented in G if and only if H*=

*N is c-supplemented in G*=

*N .*
(3)

*Let *³

*be a set of primes. Let N be a normal *³

*-subgroup of G and H a *³

*-*
*subgroup of G. If H is c-supplemented in G then H N *=

*N is c-supplemented in G*=

*N .*

Furthermore, if N normalizes H , then the converse statement also holds.
(4)

*Let L be a subgroup of G and H *≤ 8.

*L*/

*. If H is c-supplemented in G then H*
*is normal in G and H *≤ 8.

*G*/

*.*
LEMMA 2.2 Lemma 2.6]).

*Let N be a solvable normal subgroup of a group*
*G *.

*N *= 1/

*. If N *∩ 8.

*G*/ = 1

*, then the Fitting subgroup F*.

*N */

*of N is the directproduct of all minimal normal subgroups of G which are contained in N .*
Recall that a formation of groups is a class of groups
homomorphic images and is such that

*G*=

*M *∩

*N *∈
subgroups of a group

*G *with

*G*=

*M *∈
Now we let 5 be the set of all prime numbers. Then, a function

*f *defined on 5 is
called a formation function if

*f *.

*p*/, possibly empty, is a formation for all

*p *∈ 5. Achief factor

*H*=

*K *of a group

*G *is called

*f *-central in

*G *if

*G*=

*C *.

*G H *=

*K */ ∈

*f *.

*p*/ for
all prime numbers

*p *dividing |

*H*=

*K *|. A formation
Finite

*p*-nilpotent groups with some subgroups

*c*-supplemented
if there exists a formation function

*f *such that
which every chief factor of

*G *is

*f *–central in

*G*. If
a formation function

*f *, then we write
=

*L F*.

*f */ and we call

*f *a local definition
Among all the possible local definitions for a local formation
there exists exactly one of them, denoted it by

*F*, such that the formation function

*F *isboth integrated (that is,

*F*.

*p*/ ⊆
for all

*p *∈ 5) and full (that is, Æ

*p F*.

*p*/ =

*F*.

*p*/
for all

*p *∈ 5), where Æ

*p *is the class of

*p*-groups.

LEMMA 2.3 Proposition IV. 3.11]).

*Let*
*i is both an integrated and full formation function of*
*the following statements are equivalent*:
1

*p*/ ⊆

*F*2

*p*/

*for all p *∈ 5

*.*
*be a saturated formation. Assume that G is*
*a group such that G does not belong to*
*and there exists a maximal subgroup M*
*and G *=

*M F*.

*G*/

*, where F*.

*G*/

*is the Fitting subgroup.*
*Then G *=.

*G */

*is a chief factor of G, G*
*exponent p if p *> 2

*and exponent at most *4

*if p *= 2

*. Moreover, G*
*elementary abelian group or *.

*G */ =

*Z *.

*G */ = 8.

*G */

*is an elementary abeliangroup.*
**3. Main results**
We now establish our main theorems for

*p*-nilpotent groups.

THEOREM 3.1.

*Let p be the smallest prime dividing the order of a group G and P*
*a Sylow p-subgroup of G. If every minimal subgroup of P is c-supplemented in Gand when p *= 2

*, either every cyclic subgroup of order *4

*of P is also c-supplementedin G or P is quaternion-free, then G is p-nilpotent.*
PROOF. Suppose that the theorem is false and let

*G *be a counterexample of the
smallest order. Then

*G *is not

*p*-nilpotent. Since all Sylow

*p*-subgroups of

*G *areconjugate in

*G*, we see that the hypotheses of our theorem is subgroup-closure byLemma Therefore

*G *is a minimal non-

*p*-nilpotent group (that is, every propersubgroup of a group is

*p*-nilpotent but itself is not

*p*-nilpotent). By a result of Itˆo Theorem 10.3.3], we know that

*G *must be a minimal non-nilpotent group. Also by a

result of Schmidt Theorem 9.1.9 and Exercises 9.1.11], we see that

*G *is of order

*p*Þ

*q*þ, where

*q *is a prime distinct from

*p*,

*P *is normal in

*G *and any Sylow

*q*-subgroup

*Q *of

*G *is cyclic. Furthermore,

*P *is of exponent

*p *if

*p *is odd and of exponent at most
4 if

*p *= 2. Let

*A *be a minimal subgroup of

*P*. Then by our hypotheses, there existsa subgroup

*K *of

*G *such that

*G *=

*AK *and

*A *∩

*K *≤ core .

If

*A *is not normal in

*G *then

*A *∩

*K *= 1 and therefore

*K *is a maximal subgroup
of

*G *with index

*p*. Since

*p *is the smallest prime dividing the order of

*G*, we seethat

*K *is normal in

*G*. Also since

*K *is a proper subgroup of

*G*,

*K *is nilpotent. Itfollows that the Sylow

*q*-subgroup of

*K *is normal in

*G *and therefore

*G *is nilpotent,a contradiction. Hence, we may assume that every minimal subgroup of

*P *must benormal in

*G *and therefore every minimal subgroup of

*P *must be in the center of

*G*.

If

*p *is odd, then

*G *is

*p*-nilpotent by Itˆo’s lemma, a contradiction. So there remainsthe case when

*p *= 2.

Now let

*p *= 2. By the above proof, we can see that every minimal subgroup of

*P *lies in the center of

*G*. If

*P *is quaternion-free, then by applying Theorem 2.8],
1

*P */ ≤

*P *∩

*G *Æ ∩

*Z *.

*G */ = 1, where

*G *Æ is the nilpotent residual of

*G *, a
contradiction. Now let every cyclic subgroup of order 4 of

*P *be also

*c*-supplementedin

*G *and let

*B *=

*b *be a cyclic subgroup of

*P *with order 4. Then, by our hypotheses,there exists a subgroup

*K *of

*G *such that

*G *=

*B K *and

*B *∩

*K *≤ core .

lies in the center of

*G*, we may replace

*K *by

*K b*2 if necessary and we may assumethat [

*G *:

*K *] ≤ 2. If [

*G *:

*K *] = 2, then

*K *is normal in

*G *and

*K *is nilpotent. Sincethe normal

*p*-complement of

*K *is the normal

*p*-complement of

*G*,

*G *is nilpotent, acontradiction. Hence,

*K *=

*G *and

*B *must be normal in

*G*. If

*B *=

*P*, then, since

*G *is a minimal non-nilpotent group and the exponent of

*P *is at most 4, we have

*G Q*/ and therefore

*G *=

*P *×

*Q*, a contradiction. If

*P *=

*B*, then it is clear
that

*G *is

*p*-nilpotent, another contradiction. Thus, by all the above contradictions, weconclude that the theorem is true.

COROLLARY 3.2.

*Let N be a normal subgroup of a group G and p the smallest*
*prime dividing the order of G. Also let*
*be a saturated formation containing the*
*class *Æ

*p of the all p-nilpotent groups and G*=

*N *∈

*P is c-supplemented in G, and when p *= 2

*, either every cyclic subgroup of order *4

*of P is also c-supplemented in G or P is quaternion-free, then G *∈
PROOF. It is easy to see from Lemma that every minimal subgroup of

*P *is

*c*-supplemented in

*N *, and when

*p *= 2 either every cyclic subgroup of order 4 of

*P *is also

*c*-supplemented in

*N *or

*P *is quaternion-free.

*p*-nilpotent. Let

*H *be the normal

*p*-complement of

*N *. Then it is clear that

*H*
is normal in

*G *and .

*G*=

*H */=.

*N *=

*H */

Finite

*p*-nilpotent groups with some subgroups

*c*-supplemented
satisfies the hypotheses of the corollary for normal subgroup

*N *=

*H *. Now if

*H *= 1,by induction, we see that

*G*=

*H *∈
. Let

*Fi *(

*i *= 1; 2) be the full and integrated
=

*L F*.

*F */, respectively. Then, it

*F*1

*q*/ for every chief factor

*K*1

*K*2 of

*G *with

*K*1
and every prime

*q *dividing the order of |

*K *=

*F*2

*q*/ for every chief factor

*K*1

*K*2 of

*G *with

*K*1
prime

*q *dividing the order of |

*K *=
that

*H *= 1 and

*N *=

*P *is a

*p*-group. In this case, for any prime

*q *dividing the orderof

*G *with

*q *=

*p *and

*Q *∈ Syl .

*G*/, it is clear that

*P Q *is a subgroup of

*G *and hence

*P Q *is

*p*-nilpotent by Theorem and therefore we have

*P Q *=

*P *×

*Q*. It follows

*K*2 is a

*p*-group for every chief factor

*K*1

*K*2 of

*G *with

*K*1
Now by using Lemma again, we see that

*G *∈
REMARK 3.3. The hypotheses that

*p *is the smallest prime dividing the order of a
group

*G *in Theorem and Corollary cannot be removed. For example,

*G *=

*S*3,the symmetric group of order three, is an example for

*p *= 3.

THEOREM 3.4.

*Let p be the smallest prime dividing the order of a group G and P*
*a Sylow p-subgroup of G. If every maximal subgroup of P is c-supplemented in G,then G is p-nilpotent.*
PROOF. It is easy to see that every maximal subgroup of every Sylow

*p*-subgroup
of

*G *is

*c*-supplemented in

*G*. Thus, in the following proof, we may make a choiceamong Sylow

*p*-subgroups of

*G*. Now, assume that the theorem is false and let

*G*be a counterexample of minimal order. Then we prove the theorem by making thefollowing claims:

*G*/ = 1, then we may choose a minimal normal subgroup

*N *of

*G *such

*G*/. It is clear that

*P N *=

*N *is a Sylow

*p*-subgroup of

*G*=

*N *. For every
maximal subgroup

*P*1

*N *=

*N *of

*P N *=

*N *, we may assume that

*P*1 is a maximal subgroupof

*P*. Thus, by Lemma (3), every maximal subgroup of

*P N *=

*N *is

*c*-supplementedin

*G*=

*N *. Hence, by the minimality of

*G*, we know that

*G*=

*N *is

*p*-nilpotent and so

*G*is

*p*-nilpotent, a contradiction.

If

*G *is odd, then

*G *is solvable by the well-known odd order theorem of Feit and

*p G */ = 1. Now let

*G *be a group of even order and

*p G */ =

*O*2

*G */ = 1. Let

*P*1 be a maximal subgroup of

*P *. By hypotheses there
exists a subgroup

*K *of

*G *such that

*G *=

*P*
*K *= 1. Since [

*P *:

*P*1
it follows that the Sylow 2-subgroups of

*K *are cyclic of order 2 and therefore

*K *is2-nilpotent. Let

*K*2 be the Hall 2 -subgroup of

*K *. Then

*G *=

*P K*2 and

*K*2 is a
Hall 2 -subgroup of

*G*. Assume that

*G *is a non-abelian simple group. Then, by Corollary 5.6],

*G *is isomorphic to

*PSL*.2;

*r*/ with

*r *a Mersenne prime. In this case,by Corollary 5.8], every subgroup of

*G *of 2-power index is the normalizer of aSylow

*r *-subgroup of

*G*. In particular,

*K *and

*K*2 have the same order, a contradiction.

Hence

*G *is not a non-abelian simple group.

Let

*N *be a minimal normal subgroup of

*G *with

*N *=

*G*. Then

*N *is neither a
2-group nor a 2 -group. Since

*G *satisfies

*E*2 (existence of Hall 2’-subgroups), weassume that

*N*2 is a Hall 2 -subgroup of

*N *and

*N*2 a Sylow 2-subgroup of

*N *. If

*P *=

*N*2, then

*N *clearly satisfies the hypotheses of our theorem by Lemma (1).

Thus, by the minimality of

*G*, we know that

*N *is 2-nilpotent and hence

*O *.

which contradicts to (1). On the other hand, if

*N*2 is not a Hall 2 -subgroup of

*G*,then

*P N *is a proper subgroup of

*G *and

*P N *also satisfies the hypotheses of ourtheorem. Now, by the minimality of

*G *again,

*P N *is 2-nilpotent and therefore

*N *itselfis 2-nilpotent. It follows that

*O *.

*G*/ = 1, a contradiction again. Hence we conclude

*P *and

*N*2 is a Hall 2 -subgroup of

*G*. Since

*G *satisfies

*E*2 , we can see that
both

*G *and

*N *satisfy

*C*2 (all Hall 2’-subgroups are conjugate) by Gross’ theorem Main Theorem]. Now by using the Frattini argument, we have
Now let

*P*∗ ∈ Syl .

*N *.

*N *// with

*P*∗ ≤

*P*. Then, by our choice of

*G*, we know

*G*. Thus

*P *∗ <

*P *and therefore there exists a maximal subgroup

*P*1 of

*P *such that

*P*∗ ≤

*P*1. By our hypotheses again, there exists a subgroup

*K *of

*G *such that

*G *=

*P*
*K *= 1. It is now clear that the order of
Sylow 2-subgroups of

*K *is 2 and therefore

*K *is 2-nilpotent. Let

*H *be a normal2-complement of

*K *. Then,

*H *is a Hall 2 -subgroup of

*G*. Thus there exists anelement

*g *of

*G *such that

*H g *=

*N*2 . Since

*G *=

*P*1

*K *and

*H *is a normal subgroup of

*K *, we may choose

*g *∈

*P*1. We also see that

*K g *normalizes

*H g *=

*N*2 and therefore

*K g *≤

*N *.

. Thus, it follows that

*G *=

*Gg *= .

*P*1

*K */

*g *=

*P*1

*NG N*2 . This leads
to

*P *=

*P *∩

*G *=

*P *.

*p G *// = 1, then we may consider the quotient group

*G*=8.

*O p G *//. Obvi-
ously, by Lemma (2), every maximal subgroup of

*P*=8.

*O *.

*p G *//. Thus, by the minimality of

*G *, we see that

*G*=8.

*O p G *// has a nor-
mal

*p*-complement

*T *=8.

*O *.

*p G *//. By the Schur-Zassenhaus Theorem, there exists a
Hall

*p *-subgroup

*H *of

*T *such that

*T *=

*H *8.

*O *.

*p G *//. By using the Frattini argument
again, we see that

*G *= 8.

*O *.

*p G *//

*NG H */ =

*NG H */ since 8.

*O p G *// ≤ 8.

*G */, a

*p G *// = 1 and

*O p G */ is an elementary abelian group.

*p G */ is a minimal normal subgroup of

*G *.

Let

*N *be a minimal normal subgroup of

*G *such that

*N *≤

*O *.

Finite

*p*-nilpotent groups with some subgroups

*c*-supplemented
is easy to see that

*G*=

*N *satisfies the hypotheses of our theorem. The minimalityof

*G *implies that

*G*=

*N *is

*p*-nilpotent. Similarly, if

*N*1 is another minimal normalsubgroup of

*G *with

*N *≤

*Op G*/, then we see that

*G*=

*N*1 is also

*p*-nilpotent. Now

*G*=

*N *∩

*N*1 is

*p*-nilpotent, a contradiction. Hence

*N *must be the
unique minimal normal subgroup of

*G *which is contained in

*O *.

arguments similar to the proof in (3), we have

*G *=

*N N *.

*p *-subgroup of

*G*. Since

*N *.

*G H */ <

*G *, it follows that

*N *≤

*NG H */ and then, since

*p G */ ∩

*NG H */ is normal in

*G *,

*O p G */ ∩

*NG H */ = 1. Finally, by Dedekind’s law,

*p G */ =

*N *.

*O p G */ ∩

*NG H *// =

*N *. This proves (4).

(5) The final contradiction.

From the above proof, we see that

*G*=

*O *.

*p G */ is

*p*-nilpotent. By using the arguments
similar to the proof in (3), we have

*G *=

*N N *.

*G H */, where

*H *is a Hall

*p *-subgroup
of

*G*. Let

*P*∗ be a Sylow

*p*-subgroup of

*N *.

*G H */ <

*G *and therefore

*P *∗ <

*P *. Let

*P*1 be a maximal subgroup of

*P *with
Since

*Op G*/ is a minimal normal subgroup of

*G *and

*Op G*/ ≤

*P*1, we
1. By our hypotheses, there exists a subgroup

*K*1 of

*G *such that
1. It is clear that the order of Sylow

*p*-subgroups of

*K*1
is

*p *and therefore

*K*1 has a normal

*p*-complement

*H*1. Then there exists an element

*g *∈

*O *.

*p G */

*P *∗ such that .

*H*1

*H *. Noticing that

*P*1 is normal in

*Op G*/

*P*∗, we
has normal

*p*-complement and

*H *= .

*H */

*g *≤ .

and consequently

*G *=

*P *.

*P*1

*NG H */. Hence

*Op G*/

*P*∗ = .

*Op G*/

*P*∗/ ∩

*p G */

*P *∗/ ∩ .

*P*1

*NG H *// =

*P*1

*Op G*/

*P*∗/ ∩

*NG H *// =

*P*1

*P*∗ =

*P*1, which
contradicts to the fact that

*O *.

*p G */

*P *∗ is a Sylow

*p*-subgroup of

*G *. Thus our theorem
COROLLARY 3.5.

*If every maximal subgroup of every Sylow subgroup of a group*
*G is c-supplemented in G, then G is a Sylow tower group of supersolvable type.*
PROOF. Let

*p *be the smallest prime dividing the order of

*G *and

*P *a Sylow

*p*-
subgroup of

*G*. By Corollary

*G *is

*p*-nilpotent. Let

*N *be a normal

*p*-complementof

*G*. Clearly

*N *satisfies the hypotheses of

*G *and therefore by induction

*N *is a Sylowtower group of supersolvable type. This proves that

*G *is a Sylow tower group ofsupersolvable type.

By using the arguments similar to the proof in Corollary we can prove the
COROLLARY 3.6.

*Let N be a normal subgroup of a group G and p the smallest*
*prime dividing the order of G. Also let*
*be a saturated formation containing the*
*class *Æ

*p of the all p-nilpotent groups and G*=

*N *∈

*P is c-supplemented in G, then G *∈

*, where P is a Sylow p-subgroup of N .*
**4. Applications**
As an application of Theorem and Theorem we establish the following the-
orems for a group to be in the saturated formation containing the class of supersolvablegroups.

*be a saturated formation containing the class of supersolv-*
*. Let N be a normal subgroup of a group G such that G*=

*N is in*
*for every prime p dividing the order of N and for every Sylow p-subgroup P of N ,every minimal subgroup of P is c-supplemented in G and when p *= 2

*, either everycyclic subgroup of order *4

*of P is also c-supplemented in G or P is quaternion-free,then G is in*
PROOF. Assume that the theorem is false and let

*G *be a counterexample of minimal
order. By Lemma and Theorem we know that

*N *is a Sylow tower group ofsupersolvable type. Let

*q *be the largest prime dividing the order of

*N *and

*Q *aSylow

*q*-subgroup of

*N *. Then

*Q *is normal in

*G *and every minimal subgroup of

*Q*is

*c*-supplemented in

*G*. It is clear that .

*G*=

*Q*/=.

*N *=

*Q*/
satisfies the hypotheses of our theorem by Lemma The minimality of

*G *impliesthat

*G*=

*Q *is in
≤

*Q *and

*G *is a

*q*-group, where

*G *is the
-residual of

*G*. By Theorem 3.5], there exists a maximal subgroup

*M *of

*G*
such that

*G *=

*M F *.

*G*/, where

*F *.

*G*/ = Soc.

*G *mod 8.

*G*// and

*G*=core .

and therefore

*G *=

*M F*.

*G*/ since

*G*
*F*.

*G*/ is the Fitting subgroup of

*G*. It is now clear that

*M *satisfies the hypotheses of
our theorem for its normal subgroup

*M *∩

*Q*. Hence, by the minimality of

*G*, it leadsto

*M *must be in
has exponent

*q *when

*q *= 2 and exponent at most 4 when
is an elementary abelian group, then

*G*
of

*G*. For any minimal subgroup

*A *of

*G *, we know that

*A *is

*c*-supplemented in

*G*by our hypotheses. Hence there exists a subgroup

*K *of

*G *such that

*G *=

*AK *and

*G A*/. If

*A *is not normal in

*G *, then

*A *∩

*K *= 1. It is clear that

*K *∩

*G*
is normal in

*G*. The minimality of

*G*
in

*G*, a contradiction. Hence

*A *is normal in

*G *and

*G*
=

*A *is cyclic of order

*q*.

is not an elementary abelian group, then .

*G */ =

*Z *.

*G */ = 8.

*G */ is an
elementary abelian group by Lemma Noticing that 8.

*G */ ≤ 8.

*G*/, we knowthat every minimal subgroup of .

*G */ is not complemented in

*G*. It now followsfrom our hypotheses that every minimal subgroup of .

*G */ must be normal in

*G*.

Finite

*p*-nilpotent groups with some subgroups

*c*-supplemented
For any minimal subgroup

*A *of

*G *=.

*G */ , there exists a subgroup

*A *of

*G*
that

*A *=

*A*.

*G */ =.

*G */ . Assume that

*A *is of order

*q*. If

*A *is not normal in

*G*,then, by our hypotheses, there exists a subgroup

*K *of

*G *such that

*G *=

*AK *and

*A *∩

*K *= 1. Noticing that .

*G */ = 8.

*G */ ≤ 8.

*G*/, we see that

*K *=.

*G */ is a
complement of

*A*. The minimality of

*G *=.

*G */ implies that

*A *=

*G *=.

*G */ isnormal in

*G*=.

*G */ , and therefore

*G *=.

*G */ is a cyclic group of order

*q*. Hence wemay assume that

*q *= 2 and every generated element of

*G*
/ = .

*G */ = 8.

*G */ and therefore every minimal subgroup
/ is normal in

*G*. Hence 1

*G */ ≤

*Z*.

*G*/. If

*Q *is quaternion-free, then,
by Lemma 2.15], every 2 -element of

*G *acts trivially on

*G *. Since

*G *=.

*G */is a chief factor of

*G*, we see that

*G *=.

*G */ is a cyclic group of order 2. Assumethat every cyclic group of order 4 of

*Q *is

*c*-supplemented in

*G*. Let

*B *=

*b *bea cyclic group of order 4 of

*G *. Then

*b*2 is normal in

*G*. If

*B *is not normal in

*G*, then there exists a subgroup of

*K *of

*G *such that

*G *=

*B K *and

*B *∩

*K *=

*b*2 .

It is clear that .

*G */ = 8.

*G */ ≤

*K *and

*G *=.

*G */ ∩

*K *=.

*G */ is normal in

*G*=.

*G */ . The minimality of

*G *=.

*G */ implies that

*G *=.

*G */ ∩

*K *=.

*G */ = 1and therefore

*G *=.

*G */ is a cyclic group of order 2. We have now shown that for allcases,

*G *=.

*G */ is always a cyclic group of prime order. Noticing that

*G *=.

*G */is

*G*-isomorphic to Soc.

*G*=core .

*G M *//, it follows that

*G*=core

*G M */ is supersolvable,
a contradiction. Thus, our proof is completed.

*be a saturated formation containing the class of supersolv-*
*. Let N be a normal subgroup of a group G such that G*=

*N is in*
*for every prime p dividing the order of N and for every Sylow p-subgroup P of N ,every maximal subgroup of P is c-supplemented in G, then G is in*
*i i *= 1; 2/ be the full and integrated formation functions such that
=

*L F*.

*F */. Assume that the theorem is false and we may let

*G *be a minimal counterexample. Then, by applying Lemma and Corollary we know that

*N *has a Sylow tower of supersolvable type. Let

*p *be the largest primedividing the order of

*N *and

*P *∈ Syl .

*N */. Then

*P *must be a normal subgroup of

*G*. Clearly, .

*G*=

*P*/=.

*N *=

*P*/
. It is easy to see that

*G*=

*P *satisfies our
hypotheses of the theorem for the normal subgroup

*N *=

*P*. By the minimality of

*G*, wesee that

*G*=

*P *∈
, and of course, every maximal subgroup of

*P *is

*c*-supplemented
Let

*L *be a minimal normal subgroup of

*G *with

*L *≤

*P*. Then, it is easy to see
that the quotient group

*G*=

*L *satisfies the hypotheses of our theorem for the normalsubgroup of

*P*=

*L*. By our choice of

*G*, we have

*G*=

*L *∈
formation,

*L *is the unique minimal normal subgroup of

*G *which is contained in

*P*and also

*L *is complemented in

*G*. In particular, we have

*P *∩ 8.

*G*/ = 1 and therefore

*L *=

*F*.

*P*/ =

*P *is an abelian minimal normal subgroup of

*G *by Lemma
Let

*P*1 be a maximal subgroup of

*P*. By our hypotheses, there exists a subgroup

*K *of

*G *such that

*G *=

*P*
*K *= 1 since

*L *is the unique minimal normal
subgroup of

*G *contained in

*P *with

*L *≤

*P*
1. Thus

*P *=

*P*1

*P *∩

*K */. It is clear that

*P *∩

*K *is normal in

*K *and is normalized by

*P*1 since

*P *is abelian. Therefore

*P *∩

*K *is
a normal subgroup of

*G*. Since

*P *∩

*K *= 1 and

*P *is a minimal normal subgroup of

*G*,it follows that

*P *∩

*K *=

*P *and

*P *is a cyclic group of order

*p*. Since

*Aut *.

*P*) is a cyclicgroup of order

*p *− 1 and

*G*=

*C *.

*G P */ ≤ Aut.

*P */, we have

*G*=

*CG P */ ∈

*F*1

*p*/ ⊆

*F*2

*p*/,
, a contradiction. Thus, our proof is completed.

be the class of groups

*G *whose derived group

*G *is nilpotent.

is a saturated formation containing the class
by applying our Theorems and we also obtain some sufficient conditions fora group to be a

**Acknowledgement**
The authors would like to thank the referee for his comments contributed to the

**References**
[1] Z. Arad and E. Fisman, ‘On finite factorizable groups’,

*J. Algebra ***86 **(1984), 522–548.

[2] M. Asaad, A. Ballester-Bolinches and M. C. Pedraza-Aguilera, ‘A note on minimal subgroups of

finite groups’,

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-normalizers and local definitions of saturated formations of finite
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[4] A. Ballester-Bolinches, Yanming Wang and Xiuyun Guo, ‘

*c*-supplemented subgroups of finite
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[5] J. Buckley, ‘Finite groups whose minimal subgroups are normal’,

*Math. Z. ***116 **(1970), 15–17.

[6] K. Doerk and T. Hawkes,

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*M*-groups and 2-groups’,

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[8] W. Feit and J. G. Thompson, ‘Solvability of groups of odd order’,

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[10] Deyu Li and Xiuyun Guo, ‘The influence of

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*A course in the theory of groups *(Springer, New York, 1993).

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Source: http://www.austms.org.au/Publ/Jamsa/V78P3/pdf/w85.pdf

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