A note on minimal subgroups of finite groups

Abstract

All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Itô [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Itô's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is π-quasinormal in G if H permutes with every Sylow subgroup of G. More recently, Shirong [9] proved that if the finite group G possesses a normal subgroup N of odd order such that G/N is supersoluble, and if, for each Sylow subgroup P of N, every minimal subgroup of P is normal in NG(P), then G is supersoluble. In [10] and [11] Yokoyama extends the results of Itô and Buckley for soluble groups using formation theory to generalize the notion of centrality.

Full text

COMMUNICATIONS IN ALGEBRA, 24(8),2771-2776 (1996)
A note on minimal subgroups of finite groups
M.Asaad
Cairo University
Faculty oí Science
Department oí Matheqlatics
Gyza, Egypt
A. Ballester-Bolinches and M.C.Pedraza Aguilera
Departament d'Algebra. Universitat.de Valéncia
CI Doctor Moliner 50, 46100 Burjassot (Valéncia). Spain
1 Introduction and Preliminaries
All groups considered in this note will be finite. Recall that a minimal sub-
group oí a finite group is a subgroup oí prime order. Many authors have
investigated the structure oí a finite group G, under the assumption that all
mínimal subgroups o{G are well-situated in the group. Itú [7;III,5.3] proved
that ií G is a group of odd order and all mínima.l subgrollps oí G líe in the
center oí G, then Gis nilpotent. An extension oí Itú's result is the íollowing
statement [7;IV,p.435]: If for an odd prime p, every subgroup oí G oí order
p líes in the center of G, then Gis p-nilpotent. If all element oí G of orders
2 and 4 líe -in the center oí G; then G is 2-nilpotent. Buckley [4] proved
that ií Gis a group of odd order and all minimal subgroups oí G are normal
in G, then G is supersoluble. Later Shaalan [8]proved that ií G is a finite
group and every subgroup of G oí prime order or order 4 is 7r-quasinormal
in G , then G is supersoluble. Recall that a subgroup H oí a group G is
7r-quasinormal in G if H permutes with every Sylow subgro1J.poí G. More
recently, Shitong [9] proved that ií the finite group Q possesses
anormal
subgroup N oí odd order such that G/N is supersoluble, and if , íor each
Sylow subgroup P oí N, every minimal subgroup oí Pis. normal in N G(P),
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Copyright@ 1996byMareelDeldcer,me.

2772 ASAAD, BALLESTER-BOLINCHES,AND PEDRAZA AGUILERA
then G is supersoluble. In [10] and [11] Yokoyama extends the results of
Itó and Buckley for soluble groups using formation theory to generalize the
notion of centrality. .
The present note is a continuation of [3] and represents an attempt to
extend and improve the results of Shaalan and Shirong through the theory
of formations. .
Let F be a class of groups. We call F a formation provided:
(1) F contains all of homomorphic images of a group in F, and
(2) If G/M and G/N are in F, then G/(MnN) is in F for normal subgroups
M,NofG.
Each group G has a smallest normal subgroup N such that G/N is in
F. This uniquely determined normal subgroup of Gis called the F-residual
subgroup of G and \vill be denoted here by G:F. A formation F is said to
be saturated if G/iI!(G) implies G E F (see [5¡Chapter IV]).
Throughout this note U will denote the class of all supersoluble groups.
ClearIy U is a formation. Since a group Gis supersoluble if and only G/iI!(G)
supersoluble [7¡VI,p.713],it follows that U is saturated.
2 Results
Definition Let p be ~ prime and G a group, we define:
'lT(G)
=< x/x E G,Ixl is a prime number or Ix 1=4 >.
'lT*(G)=< x/x E G,Ixl is a prime number
>.
We say that an element x E G is 1r-quasinormal in G if < x > is a
1r-quasinormal subgroup of G.
Lemma 1
Let P be a minimal normal p-subgroup oi a group G, p a prime number. li
every subgroup oi P oi order p is 1r-quasinormal in G, then P is oi order p.
Proof Let D be a Sylow p-subgroup oí G. Then P
n Z(D) f; 1. Let L be
a subgroup of P n Z(D) oí order p. By the hypothesis of the lemma, L is
7r-quasinormal in G . Take now q any prime number distinct from p and Q
any Sylow q-subgroup of G. Then LQ is a subgroup of G. Moreover L is 1r-
quasinormal in LQ thus L is subnormal in LQ. Now Lis a Sylowp-subgroup
oí LQ, hence L is pronormal in LQ. Therefore L is normal in LQ by [6¡Ex-
ercise 6,p.14]. So we have that OP(G) =< Q/Q E Sylq(G), ior all q f;

-- - - - -
MINIMAL SUBGROUPSOF FINITE GROUPS 2773
p >::; NG(L). On the other hand D ::; NG(L). Therefore L is normal in G.
Since P is a minimal normal subgroup of G} we have that P = L.
Lemma 2(see Theorem 1 and Proposition 1 of [3})
Let F be a saturated formation. Assume that G is a group such that G
does not belongto F and there exists a maximal subgroup M of G such that
M E F and G
= M F(G). Then G:Fj(G:F)' is an F-eccentric chief factor
of G. G:F is a p-group for sorne prime p, G:Fhas exponent p if p
> 2 and
exponent at most 4 if p =2. Moreover, G:F is either elementary abelian or
(G:F)'
=Z( G:F) =~(G:F) is an elementary abelian group.
In [3]the second and third authors proved the following result: Let F be
a saturated íormation containing U . Suppose G is a group with a normal
subgroup H such that GjH E F. If every generator of 'I!(H) permutes
with every subgroup oí G, then G E F. This result is a consequence oí the
following theorem:
Theorem 1 Let:F be a saturated formation containing U
. Suppose G is a
group with a normal subgroup H such that GjH E F. If every generator of
W(H) is 7r-quasinormal in G, then G E F.
Proof Assume the result is false and let G be a counterexample oí mínimal
order. Then G does not belong to F and 1
=f. G:F ::; H. By [2;Theo-
rem (3.5)], G has a maximal subgrouj> M such that GjCoreG(M) does not
belong to F and G
=MF'(G) where F'(G) =Soc(G mod ()(G)). This im-
plies G =MC:F = MH and Mj(M nH) E F. Moreover every generator of
W(MnH) is 7r-quasinormalin G,so in M. Hence M satisfies the hypotheses
oí the theorem. By minimality of
G,it follows that M E F. On the other
hand, by [8;Theorems 3.3, 3.4 and 3.5], H is supersoluble . Consequently
G:Fis soluble and then G =M F( G). Applying Lemma 2, we have that G:F
is a p-group for some prime p, G:F has exponent p ií p > 2 and exponent
a.t most 4 if p
=2. Moreover G:Fj( G:F)'is a minimal normal p-subgroup of
Gj(G:F)'. It is clear that every subgroup of G:Fj(G:F)' is 7r-quasinormal in
Gj (G:F)'. By Lemma 1, we have that G:Fj (G:F)' is a cyclic group oí order
p. Since G:Fj(G:F)'is G-isomorphic to Soc(GjCoreG(M)) it follows that
GjCoreG(M) is supersoluble, a contradiction.
Remarks
(a) The above Theorem does not hold for arbitrary formations. Let F be the
íormation composed oí all groups G such that GU, the supersoluble residual,
is elementary abelian. It is clear that U
~ F but F is not saturated. Take

2774 ASAAD, BALLESTER-BOLINCHES,AND PEDRAZA AGUILERA
G
= 8L(2,3) and H = Z(G). Then G/H E F and every generator of
iJ!(H) = H is 7r-quasinormal in G. However G does not belong to F.
(b) Theorem 1 is not true for saturated formations which do not contain
the class of supersoluble groups. For example if F
=N the ~aturated for-
mation of all nilpotent groups, then the symmetric group of degree 3 has
a normal subgroupH oforder3 and I G/ H 1= 2. HoweverGis not nilpotent.'
Assad and Ramadan [1]proved the following result: Let H be a normal
subgroup of G such that G/H is supersoluble. Suppose that, for each gen-
erator x oí 'iI!(H), < x > is pronormal in G. Then G is supersoluble. In the
following Theorem we present a common generalization of that result and
the aiorementioned one of Shirong.
Theorem 2 Let F be a saturated formation containing U. Suppose that G
is a group with a normal subgroup H having a Sylow tower of supersoluble
type such that G/H E F. /f for eachSylowsubgroupP of H, everygenerator
of 'l1(P) is 7r-quasinormal in NG(P), then G belongs to F.
Proof Assume the result is false and let G be a counterexample oí minimal
order. Among the normal subgroups X oí G satisfying the hypotheses oí
the theorem, we choose H with
I H I minimal.
, Let q be the largest prime dividing I H Iand let Q be a Sylowq-subgroup
of H. It is clear that Q is anormal subgroup oí G. Consider the group G/Q.
Then H/Q is a normal subgroup of G/Q having a Sylow tower of supersol-
uble type and (G/Q)/(H/Q) E F. Now if R/Q is a Sylow p-subgroup oí
H
/Q, then p f= q and there exists a Sylow p-subgroup, P oí H such that
R =PQ. Let xQ, x E P, be a g~nerator oí 'I!(R/Q). Then x E qi(P) is
a generator oí 'l1(P). By hypothesis x is 7r-quasinormal in NG(P), So xQ
is 1r-quasinormal in NG(P)Q/Q
=NG(R/Q). Consequently G/Q satisfies
the hypotheses of the Theorem. By the minimal choice of G, it follows that
G/ Q E F and by minimality oí H; we have H
= Q. Consequently every
generator oí 'iI!(H)is 7r-quasinormal in NG(Q) =G. By Theorem 1, it fol-
lows that G E F, a contradiction.
Corollary 1 Let F be a saturated formation containing U. Let H be a
normal subgroup of a group G such that G/ H is in F. 1f for every Sylow
subgroup P of H a,ndfor every generator x 01'iI!(P), < x > is pronormal in
Na(P), then G belongs to F.
Proof It suffices to show that H has a Sylow tower oí supersoluble type
because if P is a Sylow subgroup of H andx is a generator of iJ!(P), then

MINIMAL SUBGROUPSOF FINITE GROUPS 2775
< x > is subnormal in Na(P). Therefore < x > is normal in Na(P) by
[6;Exercise 6,p.14]. In particular <.x > is 7r-quasinormal in Na(P).
Next we see that H has a Sylow tower of supersoluble type. Let p be
the smallest prime dividing I H l. We see that H is p-nilpotent applying
[9;Theorem 1]. It is clear that we may assume that p is odd. Let x be a
generator of \l1(P). Then < x > is normal in N H(P). Since Aut < x > has
order p
- 1,we have that x E Z(NH(P», So the hypotheses of Theorem 1
of [9]hold. Nowan obvious induction shows that H has a Sylowtower of
supersoluble type. Nowcorollaryfollowsfrom Theorem 2.
Similar arguments to those use in proving Corollary 1 allow us to give
the following:
Corollary 2 Under the same hypotheses as in Corollary 1, if for each Sy-
low subgroup P of H, every generator of \l1*(H) is pronormal in Na(P) and
either H is 2-nilpotent or the Sylow 2-subgroups of H are abelian, then G
belongs to :F.
The research of the sec.ondauthor is supported by Proyecto PB 94-0965
of DGICYT , MEC, Spain.
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M.Asa.a.dand M.Ramadan, On the intersection of maximal subgroups
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[2] A.Ballester-Bolinches,1í-normalizers and local definitions of saturated
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groups of finite groups.To appear in Acta Math. Hungar.
[4] J.Buckley,Finite Groups whose Minimal Subgroups are normal.
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2776 ASAAD, BALLESTER-BOLINCHES,AND PEDRAZA AGUILERA
[8] A.Shaalan, The influence oi 7r-quasinormality oi some subgroups on the
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Received: December 1995