A note on c-normal subgroups of finite groups
Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some its subgroup D satisfying 1 < |D| < |P| and study the structure of G under assumption that all subgroups H of P with |H| = |D| are c-normal in G.
Збережено в:
| Дата: | 2005 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2005
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/157201 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A note on c-normal subgroups of finite groups / A.N. Skiba // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 3. — С. 85–95. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-157201 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1572012019-06-20T01:26:17Z A note on c-normal subgroups of finite groups Skiba, A.N. Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some its subgroup D satisfying 1 < |D| < |P| and study the structure of G under assumption that all subgroups H of P with |H| = |D| are c-normal in G. 2005 Article A note on c-normal subgroups of finite groups / A.N. Skiba // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 3. — С. 85–95. — Бібліогр.: 10 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20D20. http://dspace.nbuv.gov.ua/handle/123456789/157201 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some its subgroup D satisfying 1 <
|D| < |P| and study the structure of G under assumption that all
subgroups H of P with |H| = |D| are c-normal in G. |
| format |
Article |
| author |
Skiba, A.N. |
| spellingShingle |
Skiba, A.N. A note on c-normal subgroups of finite groups Algebra and Discrete Mathematics |
| author_facet |
Skiba, A.N. |
| author_sort |
Skiba, A.N. |
| title |
A note on c-normal subgroups of finite groups |
| title_short |
A note on c-normal subgroups of finite groups |
| title_full |
A note on c-normal subgroups of finite groups |
| title_fullStr |
A note on c-normal subgroups of finite groups |
| title_full_unstemmed |
A note on c-normal subgroups of finite groups |
| title_sort |
note on c-normal subgroups of finite groups |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157201 |
| citation_txt |
A note on c-normal subgroups of finite groups / A.N. Skiba // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 3. — С. 85–95. — Бібліогр.: 10 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT skibaan anoteoncnormalsubgroupsoffinitegroups AT skibaan noteoncnormalsubgroupsoffinitegroups |
| first_indexed |
2025-07-14T09:35:34Z |
| last_indexed |
2025-07-14T09:35:34Z |
| _version_ |
1837614484373372928 |
| fulltext |
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2005). pp. 85 – 95
c© Journal “Algebra and Discrete Mathematics”
A note on c-normal subgroups of finite groups
Alexander N. Skiba
Communicated by L. A. Shemetkov
Abstract. Let G be a finite group. We fix in every non-
cyclic Sylow subgroup P of G some its subgroup D satisfying 1 <
|D| < |P | and study the structure of G under assumption that all
subgroups H of P with |H| = |D| are c-normal in G.
Introduction
Throughout this paper, all groups are finite. Following [1], we say that
a subgroup H of a group G is c-normal in G if there exists a normal
in G subgroup T such that G = HT and T ∩ H ≤ HG, where HG is
the largest normal subgroup of G contained in H. Several authors have
investigated the structure of a group G under the assumption that certain
maximal or minimal subgroups of Sylow subgroups of G are c-normal in
G. Remind, in particular, that Wang [1] proved that G is supersoluble if
either all maximal subgroups of the Sylow subgroups of G are c-normal
in G or all minimal subgroups and all cyclic subgroups with order 4 are
c-normal in G. Later on Li and Guo [2] obtained the analogous results by
limiting the c-normality of maximal or minimal subgroups to the Fitting
subgroups of a soluble groups. By using the theory of formation, Wei
[3] extended further the results to a saturated formation containing the
class of supersoluble groups. In the connection with these results the
following natural question arises: Is the group G supersoluble if for any
Sylow subgroup P of G at least one of the following conditions holds:
(1) The maximal subgroups of P are c-normal in G; (2) The minimal
2000 Mathematics Subject Classification: 20D20.
Key words and phrases: finite group, supersoluble group, c-normal subgroup,
maximal subgroup, Sylow subgroup..
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.86 A note on c-normal subgroups of finite groups...
subgroups of P and all its cyclic subgroups with order 4 are c-normal in
G? In this paper we prove the following theorem which gives the positive
answer to this question.
Theorem 0.1. Let G be a group and E be a normal subgroup of G with
supersoluble quotient G/E. Suppose that every non-cyclic Sylow subgroup
P of E has a subgroup D such that 1 < |D| < |P | and all subgroups E
of P satisfying |H| = |D| and all its cyclic subgroups with order 4 (in the
case |D| = 2) are c-normal in G. Then G is supersoluble.
As one of applications of this theorem we prove also the following
generalisation of the results from [2].
Theorem 0.2. Let G be a group and E be a soluble normal subgroup of
G with supersoluble quotient G/E. Suppose that every non-cyclic Sylow
subgroup P of F (E) has a subgroup D such that 1 < |D| < |P | and all
subgroups H of P satisfying |H| = |D| and all its cyclic subgroups with
order 4 (in the case |D| = 2) are c-normal in G. Then G is supersoluble.
1. Preliminaries
In this paper we use U to denote the class of all supersoluble groups,
ZU
∞(G) to denote the U-hypercenter of a group G that is the product
of all such normal subgroups H of G whose G-chief factors have prime
order.
Lemma 1.1. [5, II, Theorem 9.15]. G/CG(ZU
∞(G)) ∈ U .
Lemma 1.2. Let G a group and P = P1 × . . .×Pt be a p-subgroup of G
where t > 1 and P1, . . . , Pt are minimal normal subgroups of G. Assume
that P has a subgroup D such that 1 < |D| < |P | and every subroup H
of P with |H| = |D| is normal in G. Then the order of every subgroup
Pi is prime.
Proof. Assume that this is false and let G be a counterexample with
minimal order. Then for some i we have |Pi| > p and |D| > p, by
minimality of Pi. Thus if for some j we have |Pj | = p, the hypothesis is
still true for G/Pj and its subgroup P/Pj and so every G-chief factor of
between Pj and P has prime order. But then all subgroups P1, . . . , Pt are
simple, contradiction. Therefore |Pk| > p for all k = 1, . . . , t. Without
loss of generality we may suppose that D = P1 . . . Pk for some k < t. Let
M be a maximal subgroup of P1 and H = MP2 . . . PkZ where |Z| = p
and Z ≤ Pk+1. Then |H| = |D| and so by hypothesis, H is normal in G.
But then D ∩ H = MP2 . . . Pk(D ∩ Z) = MP2 . . . Pk is normal in G and
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.A. N. Skiba 87
so M = P1 ∩MP2 . . . Pk is normal in G. This contradicts the minimality
of P1.
Lemma 1.3. Let p be odd prime and P be a normal p-subgroup of a
group G. Assume that Ω1(P ) ≤ ZU (G) ∩ Z(P ). Then P ⊆ ZU (G).
Proof. Let P0 = Ω1(P ), x in G have order p2 and g ∈ G. Then (xg)p =
(xp)g = (xp)i = (xi)p for some integer i and so by [8, Theorem 1
(iv)], (xgx−i)p = 1, i.e. xg = xipu for some u ∈ P0. Thus in P/P0
every subgroup of order p is normal in G/P0. Therefore Ω1(G/P0) ⊆
ZU (G/P0) ∩ Z(P/P0) and so by induction P/P0 ⊆ ZU (G/P0). Now it
follows that P ⊆ ZU (G).
Lemma 1.4. [1, Lemma 2.1]. Let G be a group and H ≤ K ≤ G. Then
(i) If H normal in G , then H is c-normal in G.
(ii) If H is c-normal in G, the H c-normal in K.
(iii) Suppose that H is normal in G. Then K/H is c-normal in G if
and only if K is c-normal in G.
(iv) Suppose that H is normal in G. Then for every c-normal in G
subgroup T with (|H|, |T |) = 1 the subgroup HT/H is c-normal in G/H
(see [4, Lemma 2.2]).
A saturated formation is a homomorph F of groups such that each
group G has a smallest normal subgroup (denoted by GF ) whose quotient
is still in F .
Lemma 1.5. Let F be a saturated formation containing all nilpotent
groups and let G be a group with soluble F-residual P = GF . Suppose
that every maximal subgroup of G not containing P belongs to F . Then P
is a p-group for some prime p, besides, if every cyclic subgroup of P with
prime order and order 4 (if p = 2) is c-normal in G, then |P/Φ(P )| = p.
Proof. By [5, VI, Theorem 24.2], P is a p-group for some prime p and the
following hold: (1) P/Φ(P ) is a G-chief factor of P ; (2) P is a group of
exponent p or exponent 4 ( if p = 2 and P is non-abelian). Assume that
every cyclic subgroup of P with prime order and order 4 is c-normal in G.
Let X/Φ(P ) be a subgroup of P/Φ(G) with prime order, x ∈ X�Φ(P )
and L =< x >. Then |L| = p or |L| = 4 and so L is c-normal in G.
Besides, LΦ(P )/Φ(P ) = X/Φ(P ). First assume that L is not normal in
G. Then G has a normal subgroup T such that LT = G and |G : T | = p.
In this case L 6⊆ T . But on the other hand since G/T is a p-group,
L ≤ P ≤ T , by the definition of P , a contradiction. Thus L is normal in
G. But then LΦ(P )/Φ(P ) = X/Φ(P ) is normal in G/Φ(P ). Therefore
|P/Φ(P )| = p.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.88 A note on c-normal subgroups of finite groups...
Lemma 1.6. [5, II, Lemma 7.9]. Let P be a nilpotent normal subgroup
of a group G. If P ∩ Φ(G) = 1, P is the direct product of some minimal
normal subgroup of G.
Lemma 1.7. [4, III, Theorem 3.5]. Let A, B be normal subgroups of a
group G and A ≤ Φ(G). Suppose that A ≤ B and B/A is nilpotent. Then
B is nilpotent.
Lemma 1.8. [8, I, p.34]. Let p be a prime. Then the class of all p-closed
groups is a saturated formation.
Lemma 1.9. Let F be a saturated formation containing U and G be a
group with a normal subgroup E such that G/E ∈ F . If E is cyclic, then
G ∈ F .
Proof. It is enough to consider the case when E is a minimal normal
subgroup of G. Clearly E 6 subseteqΦ(G). Let M be a maximal subgroup
of G such that G = [E]M and let C = CG(E). Then MG = C ∩ M and
so G/MG = [EMG/MG](M/MG) is supersoluble, since M/MG ≃ G/C is
an abelian group. Thus G ≃ G/E ∩ MG ∈ F .
2. The proof of Theorem 0.1
Proof. Suppose that this is false and consider a counterexample for which
|G||E| is minimal.
(1) The hypothesis is still true for every Hall subgroup of E and for
G/Eπ where Eπ is any normal Hall subgroup of E (this directly follows
from Lemma 1.4).
(2) E has a non-cyclic Sylow subgroup.
Suppose that all Sylow subgroups of E are cyclic and let P be a Sylow
subgroup of E where p is the largest prime divisor of |E|. Then by [7,
IV, Theorem 2.9], E is supersoluble and so P is normal in G, as it is a
characteristic subgroup of E. Thus by (1) the hypothesis is still true for
G respectively its normal subgroup P . But by Lemma 1.9, |P | < |E|,
since G is not supersoluble, and so |P ||G| < |E||G|, a contradiction.
Now we fix some non-cyclic Sylow q-subgroup Q of E and let D be a
subgroup of Q such that 1 < |D| < |Q|, every subroup H of Q satisfying
|H| = |D| is c-normal in G and if |D| = 2, then also every subgrou with
order 4 is c-normal in G.
(3) Assume that |Q : D| = q and let N be a minimal normal subgroup
of G contained Q. Then E/N is p-closed where p is the largest prime
divisor of |E/N |. Besides, if |Q : N | 6= 4, then E/N is supersoluble.
If |Q : N | 6= 4, then by Lemma 1.4 the hypothesis is still true for E/N
and so E/N is supersoluble, by the choice of G. Assume that |Q : N | = 4.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.A. N. Skiba 89
Then either Q/N is normal in G/N or Q/N is cyclic (in these cases the
hypothesis is still true for E/N and so E/N is supersoluble) or G has a
subgroup T with |G : T | = 2 and N ≤ T . In the last case the hypotesis is
true for T/N and its normal subgroup E∩T/N , since a Sylow 2-subgroup
of E ∩ T/N has order 2. Thus a Sylow p-subgroup of E ∩ T/N is normal
in E ∩ T/N and hence E/N is p-closed.
(4) Assume that |Q : D| > q, |D| > q and all subgroups H of Q
satisfying |H| = |D| are normal in G. Suppose also that some subgroup
L of Q satisfying |L| = q is not normal in G. Then Q has at least
two different maximal subgroups which are normal in G, besides, each
non-cyclic maximal subgroup of Q is normal in G.
Let N be a minimal normal subgroup of G contained in Q. First
assume that |N | = q. Then N 6= L and so X = LN is a non-cyclic
group. Let X ≤ H ≤ K where |H| = |D| and where |K : H| = q. Then
K is not cyclic and hence it has a maximal subgroup M different from
H. Then K is normal in G as the product of normal subgroups H and
M . Analogously one can show that every subgroup of Q containing H is
normal in G. If |N | > q, N is not cyclic and as above one can show that
every subgroup of P containing N is normal in G.
(5) If |Q : D| > p, then every subgroup H of Q satisfying |H| = |D|
is normal in G.
Indeed, assume that some subgroup H of Q satisfying |H| = |D| is
not normal in G. Then G has a normal subgroup T such that DT = G
and T ∩H ≤ HG. Let M be a normal maximal subgroup of G such that
T ≤ M . Then |G : M | = q and so G/E ∩ M is supersoluble. Hence
the hypothesis is still true for G respectively its normal subgroup E∩M ,
since |Q : D| > p. But |G||E ∩ M | < |G||E|, contrary to the choice of G
and its normal subgroup E. Hence we have (4).
(6) E is soluble.
By (1) we have only to consider the case N = G. Besides, we may
assume that G is not p-closed for each prime divisor p of its order. First
assume that G is p-nilpotent for some prime divisor p of |G| and let T be
a normal p-complement of G. Since |T | < |G| and by (1), the hypothesis
is still true for T , T is supersoluble and hence G is soluble. Next suppose
that G is not p-nilpotent where p is the smallest prime divisor of |G| and
let P be a Sylow p-subgroup of G. Then P is not cyclic [9, 10.1.9] and
so by hypothesis, P has a subgroup D such that 1 < |D| < |P |, every
subroup H satisfying |H| = |D| is c-normal in G and if |D| = 2, then
also every subgroup with order 4 is c-normal in G. Besides, since G is
not p-nilpotent, it has a p-closed Schmidt subgroup (see [4, IV, Theorem
5.4]). Thus by Lemma 1.5, |D| > p. Let |P : D| = p. Suppose that
for some maximal subgroup H of P we have HG = 1 and let T be a
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.90 A note on c-normal subgroups of finite groups...
normal complement of H in G. Then the order of a Sylow p-subgroup
of T is equel to p and so the hypothesis is still true for T . Therefore T
is supersoluble, by the choice of G and hence G soluble. Hence for some
maximal subgroup H of P we have HG 6= 1. Let N be a minimal normal
subgroup of G contained in P . If |P : N | 6= 4, then G/N is supersoluble
by (3) and so G is soluble. Assume that |P : N | = 4. Then either P/N
is normal in G/N or is cyclic (in these cases the hypothesis is still true
for G/N and so G/N is supersoluble and hence G is soluble) or G has a
subgroup T with |G : T | = 2 and N ≤ T . In the last case the hypotesis
is true T/N , since a Sylow 2-subgroup of T/N has order 2. Thus T/N
is supersoluble and hence G is soluble. Finaly, assume that |P : D| > p.
Then by (5) every subgroup H of P satisfying |H| = |D| is normal in G.
Suppose that for some minimal normal subgroup N of G contained in P
we have |N | = p. If |H/N | > p, the hypothesis is true for G/N and so G
is supersoluble, since |N | = p. But this contradicts the choice of G and
hence N is a maximal subgroup of H. From above we have known that
some minimal subgroup L of P is not normal in G and so by (4), P has
at least two different maximal subgroups which are normal in G. If for
some minimal normal subgroup N of G contained in P we have |N | > p,
then by (4) every maximal subgroup M of P containing N normal in G.
Thus P is normal in G subgroup as the product of normal subgroups. It
follows that G is soluble.
(7) E is p-closed where p is the largest prime divisor of |E|.
By (1) and the choice of G we have only to consider the case E = G.
Moreover, since by Lemma 1.4 the hypothesis is still true for all Hall
subgroups of G, we may suppose that G is biprimary, i.e. |G| = paqb for
some prime q and some a, b ∈ N. Assume that G is not p-closed. Then
a Sylow q-subgroup Q of G is non-cyclic. Hence by hypothesis, Q has a
subgroup D such that 1 < |D| < |Q|, every subroup H satisfying |H| =
|D| is c-normal in G and if |D| = 2, then also every cyclic subgroup with
order 4 is c-normal in G Since G is not p-nilpotent, it has a minimal non-
p-nilpotent subgroup S, say, and by [7, IV, Theorem 5.4], S is a Schmidt
group. Hence by Lemma 1.5, |D| > p. First assume that |Q : D| = q.
Then by (2), G/N is p-closed for every minimal normal subgroup N of
G contained in Q. Thus by Lemma 1.8, N 6⊆ Φ(G) and N is the only
minimal normal subgroup of G contained in Q. By [4, III, Lemma 3.3]
for some maximal subgroup V of P we have N 6⊆ V . Let L = VG and
T be a normal subgroup of G such that V T = G and T ∩ V ≤ L. First
assume that L = 1. Then |T | = qpa and so the hypothesis is still true
for T . Therefore T is p-closed, since p is the largest prime divisor of |T |.
Hence G is closed, a contradiction. Thus L 6= 1 and so N ≤ L ≤ V , a
contradiction.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.A. N. Skiba 91
Therefore we may assume that |Q : D| > q. Then by (5) every
subgroup H of Q satisfying |H| = |D| is normal in G. If some minimal
subgroup L of Q satisfying |L| = q is not normal in G, then by (4), Q
has at least two different maximal subgroups, M and E, say, which are
normal in G. Since |Q : D| > q, the hypothesis is true for MP and EP
where P is a Sylow p-subgroup of G. But then Q = ME ≤ NG(P ) and
so G is p-closed, a contradiction. Thus all minimal subgroups of Q are
normal in G. Let N be a minimal normal subgrop of G contained in Q
and let N ≤ H where |H| = |D|. We show that G/N is p-closed. If
|N | < |D| it can be shown as above. Thus let |N | = |D|. Then since
|D| > p, N is non-cyclic and so the minimal subgroups of N are normal
non-identity subgroups of G. This contradiction completes the proof of
(7).
(8) E = Q is non-cyclic q-group.
Indeed, let p be the largest prime divisor of |E| and P be a Sylow
p-subgroup of E. Then by (7), P is normal in N and so P is normal
in G, as a characteristic subgroup of E. Besides, by (1), the hypothesis
is true for G/P and so G/P is supersoluble, by the choice of G. Thus
P = E = Q, otherwise |G||P | < |G||E|. From Lemma 1.9 it follows also
that E is not cyclic.
(9) Oq′(G) = 1.
Indeed, assume that Oq′(G) = 1 6= 1. Then |G/Oq′(G)| < |G| and so
since by Lemma 4(iv) the hypothesis is true for |G/Oq′(G)|, G/Oq′(G)
is supersoluble, by the choice of G. But then G ≃ G/(Q ∩ Oq′(G)) is
supersoluble, a contradiction.
(10) |D| > q.
Assume that |D| = q. Then since G/Q = G/E is supersoluble, GU ≤
Q and so the hypothesis is still true for G and its normal subgroup GU .
Hence GU = Q. Let M be an arbitrary maximal subgroup of G not
containing P . Then G/Q = MQ/Q ≃ M/M ∩ Q is supersoluble and
so for M and its normal subgroup M ∩ Q the hypothesis is still true.
Hence M is supersoluble, by the choice of G. Now using Lemma 1.5 we
see that Q/Φ(Q) is a chief factor of G and |Q/Φ(Q)| = q. But then, by
Lemma 1.5, G/Φ(Q) is supersoluble and so Q ≤ Φ(Q), a contradiction.
Thus we have (10).
(11) G/N is supersoluble for each minimal normal subgroup N of G
contained in Q.
We have only to show that the hypothesis is still true for G/N . By
(8) it is clear that the hypothesis is true for G/N if either Q : D| = q or
|N | = q > 2. Therefore we may suppose that |Q : D| > q and that either
|N | = 2 or N is not cyclic. By (4) every subgroup H of P satisfying
|H| = |D| is normal in G. If N is non-cyclic we conclude from (4) that
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.92 A note on c-normal subgroups of finite groups...
each maximal subgroup of Q contaning N is normal in G and we again
see that the hypothesis is true for G/N . So let |N | = 2. In this case
the hypothesis is true for G/N if Q is a Sylow q-subgroup of G, so let Q
be a proper subgroup of Sylow q-subgroup of G. Then G has a normal
maximal subgroup M such that Q ≤ M . Since the hypothesis is still true
for M , M is supersoluble and so a Sylow p-subgroup Mp is normal in G
where p is the largest prime divisor of |M |. But this Mp ≤ Oq′(G) and
so Oq′(G) 6= 1, contrary to (9).
(12) Final contradiction.
Let N be a minimal normal subgroup of G contained in Q. Then by
(9), N is the only minimal normal subgroup of G contained in Q and
N 6⊆ Phi(G). Let M be a maximal subgroup of G such that N 6⊆ M
and let C = CG(N). Then G = [R]M and Q ⊆ C, by [5, I, Corollary
4.1.1]. Hence Q ∩ M is normal in G. But Q = Q ∩ NM = N(Q ∩ M)
and so Q = N . Since Q is not cyclic, it has such a subgroup D that
1 < |D| < |P | and every subgroup H satisfying |D| = |H| is c-normal in
G. But it is impossible because P is a minimal normal subgroup of G.
This contradiction completes the proof of this theorem.
3. Some corollaries of Theorem 0.1
In this section we consider some applications of Theorem 0.1.
As an immediate consequence of Theorem 0.1, we have:
Corollary 1. Let G be a group. Suppose that for any non-cyclic Sylow
subgroup P of G at least one of the following conditions holds:
(1) The minimal subgroups of P and all its cyclic subgroups with order
4 are normal in G;
(2) The maximal subgroups of P are normal in G.
Then G is supersoluble.
Corollary 2. (Buckley [8]) Let G be a group of odd order. If all subgroups
of G of prime order are normal in G, then G is supersoluble.
Corollary 3. (Srinivasan [9]). If the maximal subgroups of the Sylow
subgroups of G are normal in G, then G is supersoluble.
Corollary 4. Let G be a group. Suppose that for any non-cyclic Sylow
subgroup P of G at least one of the following conditions holds:
(1) The minimal subgroups of P and all its cyclic subgroups with order
4 are c-normal in G;
(2) The maximal subgroups of P are c-normal in G.
Then G is supersoluble.
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.A. N. Skiba 93
Corollary 5. (Wang [1]).If all subgroups of G of prime order or order 4
are c-normal in G, then G is supersoluble.
Corollary 6. (Wang [1]).If the maximal subgroups of the Sylow subgroups
of G are c-normal in G, then G is supersoluble.
Now using Theorem 0.1 we prove Theorem 0.2.
Proof. Assume that this is false and let G be a counterexample with
minimal |G||E|. Let F = F (E) and p be the largest prime divisor of
|F |. Let P be the Sylow p-subgroup of F , P0 = Ω1(P ) and C = CG(P0).
Clearly C is normal in G.
(1) The hypothesis is true for E and for every normal subgroup of G
having coprime order with |E| (this directly follows from Lemma 1.4).
(2) p > 2.
Indeed, suppose that p = 2. Assume that E 6= G. Then E is su-
persoluble, by (1) and the choice of G. Hence a Sylow p-subgroup Ep
of E is normal in E where p is the largest prime divisor of E. Hence
Ep = P = E. But in this case G is supersoluble, by Theorem 0.1, a
contradiction. Therefore E = G is a soluble group and hence CG(F ) ≤ F
is a 2-group, by [5, II, Theorem 7.12]. Let Q be a subgroup of G with
prime order q where q 6= 2 and let X = FQ. Then the hypothesis is still
true for X and so it is supersoluble, by the choice of G. But then Q is
normal in X and so Q ≤ CG(F ), a contradiction. Hence we have (2).
(3) P0 6⊆ ZU
∞(G) ∩ Z(P ).
Suppose that P0 ≤ ZU
∞(G) ∩ Z(P ). We show that the hypothesis is
true for G/P0 and its normal subgroup CE/P0 where CE = C ∩ E. .
By Lemma 1.1, G/C is supersoluble and hence G/CE is supersoluble,
since G/E is supersoluble, by hypothesis. Clearly F = F (CE) and so
since P0 ≤ Z(CE), F (CE/P0) = F/P0. Now making use Lemma 1.3
we see that the hypothesis is still true for G/P0 and so this quotient is
supersoluble, by the choice of G. Since P0 ≤ ZU
∞(G), it follows that G is
supersoluble, by Lemma 1.9, a contradiction. Thus we have (3).
By (3), P is not cyclic and so by hypothesis P has a subgroup D
such that 1 < |D| < |P | and every subgroup H of P with |H| = |D| is
c-normal in G.
(4) |D| > p.
Suppose tat |D| = p. By (3), P has a subgroup H with prime order
which is not normal in G. Let T be a normal complement of H in
G. Then the hypothesis is true for G and its subgroup V = T ∩ E.
Indeed, evidently G/T is supersoluble and F (V ) ≤ F (E). On the other
hand, since |G : T | = p, every Sylow p-subgroup of F where q 6= p is
contained in T . Thus the hypothesis is still true for G and its subgroup
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.94 A note on c-normal subgroups of finite groups...
V , by Lemma 1.4. But since T is a proper subgroup of G and evidently
ET = G, |T | < |E|, which contradicts the choice of G and its normal
subgroup E. This contradiction completes the proof of (4).
(5) If L is a minimal normal subgroup of G and L ≤ P , |L| > p.
Assume that |L| = p. Let C0 = CG(L). Then the hypothesis is
true for G/L and its subgroup C0/L. Indeed, since L ≤ Z(C0), we have
F (C0/L) = F/L. On the other hand, if H/L is a subgroup of G/L such
that |H| = |D|, we have 1 < |H/L| < |P/L|, by (4). Besides, H/L is
c-normal in G/L we have only to use Lemma1.4(iv).
(6) Φ(G) ∩ P = 1.
Suppose that Φ(G)∩P 6= 1 and let L be a minimal normal subgroup
of G contaned in Φ(G) ∩ P . Then by (5), L is non-cyclic and so every
subgroup of G, containing L is non-cyclic. Clearly |L| ≤ |D|. We show
that the hypothesis is true for G/L and its normal subgroup E/L. First
of all note that by Lemma 1.8, F (E/L) = F/L. By Lemma 1.4(iv) we
may assume that |P : L| > p. Let |L| = |D| and let L ≤ K, M ≤ K
where M 6= L and L, M are maximal subgroups of K. We have only
to show that K is c-normal in G. It is evident if M is normal in G.
Let L = KG and T be a normal subgroup of G such that MT = G
and T ∩ M ≤ MG 6= M . Let S be a normal subgroup of G such that
|G : S| = p and T ≤ S. Then evidently KS = G and S ∩ K ≤ KG = L,
since L ≤ Φ(G) ≤ S. Therefore K is c-normal in G. Thus the hypothesis
is still true for G/L and hence G/L is supersoluble, by the choice of G.
Since the class supersoluble groups is a saturated formation, it follows
that G is supersoluble, a contradiction.
(7) P is the direct product of some minimal normal subgroups of G
(this directly follows from (6) and Lemma 1.6).
(8) Final contradiction.
If every subgroup H of G with |H| = |D| is normal in G, then by
(7) and Lemma 1.3, P ≤ ZU
∞(G), which contradicts (3). Thus for some
of such subgroups H we have H 6= HG and so G has a normal maximal
subgroup T such that PT = G. In this case a chief factor P/P ∩ T of G
has prime order and so by (7) for some minimal normal subgroup L of G
contained in P we have |P | = p, which contradicts (5).
The following corollaries are consequences of Theorem 0.2.
Corollary 7. Let G a group and E a soluble normal subgroup of G with
supersoluble quotient G/E. Suppose that for any non-cyclic Sylow sub-
group P of F (E) at least one of the following conditions holds:
(1) The minimal subgroups of P and all its cyclic subgroups with order
4 are c-normal in G;
Jo
u
rn
al
A
lg
eb
ra
D
is
cr
et
e
M
at
h
.A. N. Skiba 95
(2) The maximal subgroups of P are c-normal in G.
Then G is supersoluble.
Corollary 8. (Li and Guo [2]). Let G a group and E a soluble normal
subgroup of G with supersoluble quotient G/E. If all maximal subgroups
of the Sylow subgroups of F (E) are c-normal in G, then G is supersoluble.
Corollary 9. (Li and Guo [2]). Let G a group and E a soluble normal
subgroup of G with supersoluble quotient G/E. If all subgroups of F (E)
of prime order or order 4 are c-normal in G, then G is supersoluble.
References
[1] Wang Y., c-normality of groups and its properties, J. Algebra, 180, 1996, pp.954-
965.
[2] Li D., Guo X., The influence of c-normality of subgroups on the structure of finite
groups, II. Comm. Algebra, 26, 1998, pp.1913-1922.
[3] Wei H. On c-normal maximal and minimal subgroups of Sylow subgroups of finite
groupf, Comm. Algebra, 29, 2001, pp.2193-2200.
[4] Huppert B., Endliche Gruppen I, Springer-Verlag. Heidelberg/New York, 1967.
[5] Shemetkov L.A., Formation of finite groups, Nauka, Main Editorial Board for
Physical and Mathematical Literature, Moscow, 1978.
[6] Doerk K. and Hawkes T., Finite soluble grousp, Walter de gruyter, Berlin/New
York, 1992.
[7] Robinson D.J.S., A Course in the Thery of Groups, Springer-Verlag, New
York/Heidelberg/Berlin, 1982.
[8] Buckley J. Finite groups whose minimal subgroups are normal Math. Z., 116,
1970, pp.15—17.
[9] Srinivasan S. Two sufficient conditions for supersolubility of finite groups, Israel
J. Math., 35, 1980, pp.210-214.
[10] Robinson D.J.S., A Course in the Thery of Groups, Springer-Verlag, New
York/Heidelberg/Berlin, 1982.
Contact information
A. N. Skiba Gomel State University of F.Skorina, Be-
larus, 246019, Gomel, Sovetskaya Str., 104
E-Mail: skiba@gsu.unibel.by
Received by the editors: 14.06.2005
and final form in 15.09.2005.
|