On c-normal and hypercentrally embeded subgroups of finite groups
In this article, we investigate the structure of a finite group G under the assumption that some subgroups of G are c-normal in G
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Інститут прикладної математики і механіки НАН України
2015
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| Cite this: | On c-normal and hypercentrally embeded subgroups of finite groups / Ning Su, Yanming Wang // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 270–282. — Бібліогр.: 13 назв. — англ. |
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| author | Ning Su Yanming Wang |
| author_facet | Ning Su Yanming Wang |
| citation_txt | On c-normal and hypercentrally embeded subgroups of finite groups / Ning Su, Yanming Wang // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 270–282. — Бібліогр.: 13 назв. — англ. |
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| container_title | Algebra and Discrete Mathematics |
| description | In this article, we investigate the structure of a
finite group G under the assumption that some subgroups of G are
c-normal in G
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 2, pp. 270–282
© Journal “Algebra and Discrete Mathematics”
On c-normal and hypercentrally embeded
subgroups of finite groups∗
Ning Su and Yanming Wang
Communicated by L. A. Kurdachenko
Abstract. In this article, we investigate the structure of a
finite group G under the assumption that some subgroups of G are
c-normal in G. The main theorem is as follows:
Theorem A. Let E be a normal finite group of G. If all sub-
groups of Ep with order dp and 2dp (if p = 2 and Ep is not an
abelian nor quaternion free 2-group) are c-normal in G, then E is
p-hypercyclically embedded in G.
We give some applications of the theorem and generalize some
known results.
1. Introduction
All groups considered in this paper are finite. We use conventional
notions and notation, as in [3]. G always denotes a finite group, |G| the
order of G, π(G) the set of all primes dividing |G|, Gp a Sylow p-subgroup
of G for any prime p ∈ π(G).
A well know result is that G is nilpotent if and only if every maximal
subgroup of G is normal in G. In [11], Wang defined c-normality of a
subgroup and prove that a finite group G is solvable if and only if every
maximal subgroup of G is c-normal in G.
∗The research has been supported by NSF China (11171353).
2010 MSC: 20D10.
Key words and phrases: c-normal, hypercenter, p-supersolvable, p-nilpotent.
N. Su, Y. Wang 271
Definition 1.1 ([11], Definition 1.1). Let G be a group. We call a sub-
group H is c-normal in G if there exit a normal subgroup N of G such
that HN = G and H ∩ N 6 HG.
The basic properties of c-normality are as follows.
Lemma 1.2 ([11], Lemma 2.1). Let G be a group. Then
(1) If H is normal in G , then H is c-normal in G.
(2) G is c-simple if and only if G is simple.
(3) If H is c-normal in G , H 6 K 6 G, then H is c-normal in K.
(4) Let K E G and K 6 H, Then H is c-normal in G if and only if
H/K-normal in G/K.
Several authors successfully use the c-normal property of some p-
subgroups of G to determine the structure of G. (see [2],[5], [8-10]). Many
results in previous papers have the following form: Suppose that G/E is
supersolvable (or G/E ∈ F , where F is a formation containing the class of
all supersolvable groups), if some subgroups of E with prime power order
are c-normal G , then G is supersolvable (or G ∈ F). Actually, in a more
general case, if we can get a criterion that E lies in the F-hypercenter,
then G/E ∈ F implies that G ∈ F . In order to get good results, many
authors have to impose the c-normal hypotheses on all the prime divisors
or the minimal or maximal divisor p of |G| rather than any prime divisor.
Let p be a fixed prime. In this paper, we mainly focus on how a
normal subgroup E has the above property provided every p-subgroup of
E with some fix order is c-normal in G. For this purpose, we introduce
the concept of p-hypercentrally embeded:
Definition 1.3. A normal subgroup E is said to be p-hypercentrally
embedded in G if every p-chief factor of G below E is cyclic.
It is of a lot interest to determine the structure of G with hypothesis
that some p-subgroups are well suited in G. Many results on minimal
p-subgroups and maximal subgroups of Sylow subgroups were obtained.
Recently, people have more interest to get unified and general results
([8],[12]). That is, to consider the p-subgroups with the same order. For
simplicity, we give the following notation of dp .
Let E be a normal finite group of G. dp is a prime power divisor of
|Ep| satisfying the following properties:If |Ep| = p then dp = |Ep| = p; if
|Ep| > p then 1 < dp < |Ep|.
In this paper, we will prove the following theorem:
272 c-normal subgroups
Theorem A. Let E be a normal finite group of G. If all subgroups of Ep
with order dp and 2dp (if p = 2 and Ep is not an abelian nor quaternion free
2-group) are c-normal in G, then E is p-hypercyclically embedded in G.
As an application of Theorem A, we have the following:
Theorem B. Let E be a normal finite group of G such that both NG(Ep)
and G/E are p-nilpotent. If either Ep is abelian or every subgroup of Ep
with order dp (dp is a prime power divisor of |Ep| and 1 < dp < |Ep|) and
2dp (if p = 2 and Ep is not quaternion free) is c-normal in E, then G is
p-nilpotent.
2. Proof of the theorems
In this section, we will investigate how a normal subgroup E embedded
in G if, for a fixed prime p, some subgroups of Ep are c-normal in G. First,
we need some results about a normal subgroup with some subgroups
being c-supplemented in G. Following [10], a group H is said to be c-
supplemented in G if there exists a subgroup K of G such that G = HK
and H ∩ K 6 HG. It is clear from the definition that if a subgroup H is
c-normal in G, then H is c-supplemented in G.
Lemma 2.1. If N is a minimal abelian normal subgroup of G then all
proper subgroups of N are not c-supplement in G.
Proof. Suppose this Lemma is not true and let H be a proper subgroup of
N which is c-supplemented in G. Obviously HG = 1 since HG < N and N
is a minimal normal subgroup of G. By the definition of c-supplement, there
exit a proper subgroup M of G such that G = HM with H ∩M 6 HG = 1.
Hence NM > HM = G. Since N is abelian, we know that N ∩ M E G.
Hence N ∩ M = 1. Therefore we have |G| = |NM | = |N ||M | > |H||M | =
|HM |, a contradiction to G = HM .
For a saturated formation F , the F-hypercenter of a group G is
denoted by ZF (G) (see [3, p 389, Notation and Definitions 6.8(b)]). Let
U denote the class of all supersolvable groups. In [2], Asaad gave the
following result: Let p be a nontrivial normal p-subgroup, where p is an
odd prime, if every minimal subgroup of P is c-supplemented in G, then
P 6 ZU(G). It is helpful to give a result for p = 2. In fact, we have the
following property:
Property 2.2. Let P be a normal 2-subgroup of G. If all minimal sub-
groups of P and all cyclic subgroups of P with order 4 (if P is neither
abelian nor quaternion free) are c-supplemented in G, then P 6 Z∞(G).
N. Su, Y. Wang 273
Proof. Let Q be a Sylow q-subgroup of G (q 6= p), we are going
to show that PQ is 2-nilpotent. Suppose PQ is not 2-nilpotent,
then PQ contains a minimal non 2-nilpotent subgroup H. By Ito’s
famous result, we know that H = [H2]Hq, exp(H2) 6 4 and H2/Φ(H2)
is a minimal normal subgroup of H/Φ(H2). If |H2/Φ(H2)| = 2,
then we have |H/Φ(H2) : HqΦ(H2)/Φ(H2)| = |H2/Φ(H2)| = 2 and thus
HqΦ(H2)/Φ(H2) is normal in H/Φ(H2), which will lead to the nilpotent
of H. Therefore |H2/Φ(H2)| > 2. We distinguish the three cases:
Case 1. Every minimal subgroup of P and every cyclic subgroups
with order 4 of P is c-supplemented in G. Let 〈x〉 be a subgroup
of H2 not contained In Φ(H2), then 〈x〉Φ(H2)/Φ(H2) is a nontrivial
subgroup of H/Φ(H2). Since exp(H2) 6 4, we know that 〈x〉 is
c-supplemented in G and thus c-supplemented in H by [10, Lemma 2.1(1)].
By Lemma 2.1, we have 〈x〉Φ(H2)/Φ(H2) = H2/Φ(H2). But then
|H/Φ(H2) : HqΦ(H2)/Φ(H2)| = |〈x〉Φ(H2)/Φ(H2)| = 2, a contradiction.
Case 2. Every minimal subgroup of P is c-supplemented in G and P is
an abelian 2-group. Let 〈x〉 be a subgroup of H2 not contained In Φ(H2).
If |x| = 2, then we can get a contradiction by using exactly the same
argument as we did in Case 1.Therefore we may assume that Ω1(H2) 6
Φ(H2), where Ω1(H2) is a subgroup generated by all minimal subgroup
of H2. Since H is a minimal non 2-nilpotent group and Φ(H2)Hq < H,
Φ(H2)Hq is a nilpotent group. As a result, Hq acts trivially on Ω1(H2).
Note that H2 is also an abelian 2-group, by [4, Theorem2. 4] Hq also acts
trivially on H2, a contradiction.
Case 3. Every minimal subgroup of P is c-supplemented in G and P is a
non-abelian quaternion free 2-group. If H2 is abelian, then we can get the
same contradiction as Case 2. Hence we may assume that H2 is also a
non-abelian quaternion free 2-group. Applying [6, Theorem 2.7], Hq acts
on H2/Φ(H2) with at least one fixed point. Bare in mind that H2/Φ(H2)
is a minimal normal subgroup of H/Φ(H2), we have |H2/Φ(H2)| = 2,
again a contradiction.
The above proof shows that PQ is 2-nilpotent and thus Q E PQ.
Note that P is a normal subgroup of G, we have [P, Q] = 1. Note that
we can choose Q to be a Sylow q-subgroup of G for any q 6= p, we have
[P, O2(G)] = 1. Let H/K be a G-chief factor of P . The fact [P, O2′
(G)] = 1
yields that G/CG(H/K) is a 2-group. But by [3, A, Lemma 13.6], we have
O2(G/CG(H/K)) = 1. Consequently G/CG(H/K) = 1 for any G-chief
factor of P , in other words, P 6 Z∞(G).
As an application of Property 2.2, we have:
274 c-normal subgroups
Corollary 2.3. If all minimal subgroups of G2 and all cyclic subgroups
of G2 with order 4 (if G2 is neither abelian nor quaternion free) are
c-supplemented in G, then G is 2-nilpotent.
Proof. Suppose this corollary is not true and let G be a counterexample
with minimal order. Obviously the hypothesis is inhered by all subgroups
of G, G is actually a minimal non 2-nilpotent group. Hence G2 is a normal
subgroup in G. Applying Property 2.2 to G2, we get a contradiction.
By combining [2, Theorem 1.1] and Property 2.2, we have:
Lemma 2.4. Let P be a normal p-subgroup of G. If all cyclic subgroups
of P with order p or 4 (if P is a non-abelian and not quaternion free
2-group) are c-supplement in G, then P 6 ZU (G).
Next, we will show that if that some class of p-subgroup is c-normal
in G, then G is p-solvable.
Lemma 2.5. If Gp is c-normal in G then G is p-solvable.
Proof. Suppose this Lemma is not true and considered G to be a counterex-
ample with minimal order. Clearly the hypothesis holds for any quotient
group of G, the minimal choice of G implies that Op(G) = Op′(G) = 1.
By the definition of c-normal, there exit a normal subgroup H of G such
that G = GpH and H ∩ Gp 6 (Gp)G. But (Gp)G = Op(G) = 1, hence H
is a p′ normal subgroup of G. The fact Op′(G) = 1 indicates that H = 1
and thus G = Gp, a contradiction.
Lemma 2.6. Let dp be a prime power divisor of |Gp| with dp > 1. If
every subgroup of |Gp| with order dp and 2dp (If p = 2 and G2 is neither
abelian nor quaternion free)) is c-normal in G then G is p-solvable.
Proof. Suppose this Lemma is not true and considered G to be a coun-
terexample with minimal order. According to Lemma 2.5 we may assume
that 1 < dp < |Gp|.
(1) Op′(G) = 1.
Since the hypothesis holds for G/Op′(G), the minimal choice of G
yields that Op′(G) = 1.
(2) Every subgroup with order dp and 2dp (if p = 2 and G2 is neither
abelian nor quaternion free) is normal in G. In particular, Op(G) > 1.
Suppose there exit a subgroup K with order dp or 2dp (if p = 2)
that is not normal in G. Then there exit a proper normal subgroup
N. Su, Y. Wang 275
L such that G = KL, K ∩ L 6 KG. Since G/L is a p-group, we can
find a normal subgroup M containing L such that |G/M | = p. But
dp < |Gp| so M still satisfies the hypothesis of this Lemma, thus M
is p-solvable by the minimal choice of G and so is G.
(3) Let N is a minimal normal subgroup contained in Op(G), then
|N | = dp.
If dp > |N |, then G/N satisfies the hypotheses of this Lemma and
thus is p-solvable by the minimal choice of G. Since N is a p-group
we can get that G is p-solvable, a contradiction.
(4) dp = p.
Suppose dp > p. From (3) we know that |N | = dp > p and thus N
is not cyclic. Let H be a subgroup of Gp containing N such that
|H : N | = p. Let M1 and M2 be two different maximal subgroup
of H. By (2), both M1 and M2 are normal in G. Consequently
H/N = M1M2/N is also normal in G/N . Hence every subgroup of
G/N with order p is normal in G/N . If p = 2 and and G2 is neither
abelian nor quaternion free, then by using a similar argument we
know that every subgroup of G/N with order 4 is also normal in
G/N . As a result, we see that G/N satisfies the hypothesis of this
Lemma and the choice of G implies that G/N is p-solvable, thus G
is p-solvable, a contradiction.
(5) Final contradiction.
If p = 2, then from (4) and Corollary 2.3, G is 2-nilpotent. So we
may assume p is an odd prime. By (2) and (4) we know that every
subgroup with order p is normal. Take a subgroup 〈x〉 with order p ,
it’s easy to see that Gp 6 CG〈x〉. If CG〈x〉 < G then from the choice
of G we know that CG〈x〉 < G is p-solvable. But G/CG〈x〉 < G
is cyclic and thus G is p-solvable, contradict to the choice of G.
Therefore we have CG〈x〉 = G, that is, every minimal subgroup of
order p is contained Z(G). From Ito’s theorem G is p-nilpotent, a
contradiction.
Now, we will study the properties of p-hypercyclically embedding. In
[7, p. 217], a normal subgroup E is said to be hypercyclically embedded
in G if every chief factor of G below E is cyclic. If a normal subgroup E
is hypercyclically (p-hypercyclically) embedded in G, then E is solvable
(p-solvable) and every normal subgroup of G contained in E is also
hypercyclically (p-hypercyclically) embedded in G. The following lemma
shows that for a p-solvable normal subgroup E, we can deduce that E
276 c-normal subgroups
is hypercyclically (p-hypercyclically) embedded in G from the maximal
p-nilpotent normal subgroup of E Fp(E).
Lemma 2.7. A p-solvable normal subgroup E is hypercyclically (p-
hypercyclically) embedded in G if and only if Fp(E) is hypercyclically
(p-hypercyclically) embedded in G. In particular, if E is a p-solvable nor-
mal subgroup with Op′(E) = 1, then E is hypercyclically embedded in G
if and only if Op(E) is hypercyclically embedded in G.
Proof. We only need to prove the sufficiency. Suppose the assertion is
false and let (G, E) be a counterexample with |G||E| minimal. We claim
that Op′(E) = 1. Indeed, since Fp(E/Op′(E)) = Fp(E)/Op′(E), it’s easy
to verify that the hypothesis still holds for (G/Op′(E), E/Op′(E)). If
Op′(E) 6= 1, then the the minimal choice of (G, E) implies that E/Op′(E)
hypercyclically (or p-hypercyclically) embedded in G/Op′(E). Since we
have that Op′(E) is a normal subgroup of G contained in E, Op′(E) is
hypercyclically (or p-hypercyclically) embedded in G . Therefore we have
E hypercyclically (or p-hypercyclically) embedded in G, a contradiction.
Let N be a minimal normal subgroup of G contained in E. N is
an abelian normal p-subgroup since E is p-solvable and Op′(E) = 1.
Consider the group CE(N)/N . Let L/N = Op′(CE(N)/N) and K be
the Hall p′ subgroup of L. Then L = KN . Since K 6 L 6 CE(N), we
have K = Op′(L) 6 Op′(G) = 1. Consequently Op′(CE(N)/N) = 1 and
we have Fp(CE(N)/N) = Op(CE(N)/N) 6 Op(E)/N = Fp(E)/N . As
a result, we know that the hypothesis holds for (G/N, CE(N)/N) and
the minimal choice of (G, E) yields that CE(N)/N is hypercyclically (or
respectively p-hypercyclically) embedded in G/N . But N 6 Fp(G) and
thus N is also hypercyclically (or p-hypercyclically) embedded in G. Thus
CE(N) is hypercyclically (or p-hypercyclically) embedded in G.
Since N is a normal p-subgroup which is hypercyclically (or respec-
tively p-hypercyclically) embedded in G, we have that |N | = p. It yields
G/CG(N) is a cyclic group. As a result, ECG(N)/CG(N) is hypercycli-
cally embedded in G/CG(N). Note that E/CE(N) = E/E ∩ CG(N) is
G-isomorphic with ECG(N)/CG(N), therefore E/CE(N) is hypercycli-
cally embedded in G/CE(N). But CE(N) is also hypercyclically (or
p-hypercyclically) embedded in G hypercyclically (or p-hypercyclically)
embedded in G and thus E is hypercyclically (or p-hypercyclically) em-
bedded in G, a final contradiction.
Denote A(p − 1) as the formation of all abelian groups of exponent
divisible by p − 1. The following proposition is well known:
N. Su, Y. Wang 277
Lemma 2.8 ([12], Theorem 1.4). Let H/K be a chief factor of G, p is a
prime divisor of |H/K|, then |H/K| = p if and only if G/CG(H/K) ∈
A(p − 1).
Let f be a formation function, and N be a normal subgroup of G.
We say that G acts f -centrally on E if G/CG(H/K) ∈ f(p) for every
chief factor H/K of G below E and every prime p dividing |H/K| ([3],
p. 387, Definitions 6.2). Fixing a prime p, define a formation function gp
as follows:
gp(q) =
{
A(p − 1) (if q = p)
all finite group (if q 6= p)
From Lemma 2.8, we can see that E is p-hypercyclically embedded
in G if and only if G acts gp-centrally on E. By applying [3, p. 388,
Theorem 6. 7], we get the following useful results:
Lemma 2.9. A normal subgroup E of G is p-hypercyclically embedded
in G if and only if E/Φ(E) is p-hypercyclically embedded in G/Φ(E).
Lemma 2.10. Let K and L be two normal subgroup of G contained in E.
If E/K is p-hypercyclically embedded in G/K and E/L is p-hypercyclically
embedded in G/L, then E/L∩K is p-hypercyclically embedded in G/L∩K.
The following proposition indicates that when dp = p, the conclusion
of Theorem A holds.
Proposition 2.11. Let E be a normal subgroup of G. If all cyclic sub-
groups of Ep with order p and 4 (if p = 2 and Ep is not an abelian nor
quaternion free 2-group) are c-normal in G, then E is p-hypercyclically
embedded in G.
Proof. Suppose this Theorem is not true and let (G, E) be a counterex-
ample such that |G| + |P | is minimal. Suppose Op′(E) 6= 1, it’s easy to
verifies that (G/O′
p(E), E/O′
p(E)) satisfies the hypothesis of this Theorem
and thus E/O′
p(E) is p-hypercyclically embedded in G/Op′(E) by the
minimal choice of (G, E). But then E is p-hypercyclically embedded in
G. This contradiction implies that Op′(E) = 1.
From Lemma 2.6 and Lemma 1.2(3) we know that E is p-solvable
and from Corollary F we know that Op(E) 6 ZU (G), thus E 6 ZU (G) by
Lemma 2.7, a contradiction.
With the aid of all the preceding results, we can now prove the main
theorem of this section.
278 c-normal subgroups
Proof of Theorem A. Suppose this is not true and let (G, E) be a
counterexample such that |G| + |E| is minimal. If |Ep| = p, then Ep
itself is c-normal in G and by Lemma 1.2, Ep is also c-normal in E.
By Lemma 2.5 we know that E is p-solvable and consequently E is p-
hypercyclically embedded in G since |Ep| = p. Therefore we may assume
that |Ep| > p and 1 < dp < |Ep|. By Proposition 2.11, we may further
assume that dp > p. Similar to step (1) in the proof of Lemma 2.6, we
have Op′(E) = 1. By Lemma 2.6, E is p-solvable. Let N be a minimal
normal subgroup of G contained in E, then obviously N 6 Op(E).
(1) |N | > p.
Suppose |N | = p, then dp > |N | by our assumption that dp > p.
Hence (G/N, E/N) also satisfies the hypothesis of this Theorem
and therefore E/N is p-hypercyclically embedded in G/N by the
choice of (G, E). If |N | = p, then E is p-hypercyclically embedded
in G, a contradiction.
(2) dp > |N |.
By Lemma 2.1 we have dp > |N |. Suppose that dp = |N |. Since
dp < |Ep| by our assumption, let H be a subgroup of Ep such that
N is a maximal subgroup of H . By (1), N is not cyclic and so is H.
Hence we can choose a maximal subgroup K of H other than N .
Obviously we have H = NK. If N ∩K = 1, then |N | = |H|/|K| = p,
contradict to (1). Thus N ∩ K 6= 1 and |K : K ∩ N | = |KN : N | =
|H : N | = p. Since KG ∩ N 6 K ∩ N < N , we have KG ∩ N = 1. If
KG 6= 1, then H = NKG and K = K ∩ KGN = (K ∩ N)KG. As a
result, |KG| = |K|/|K ∩ N | = p. But this contradicts to (1) because
now we find a normal subgroup of G contained in Op(G) with order p.
Therefore we have KG = 1. Since |K| = |N | = dp, K is c-normal in
G by the hypothesis of this theorem. So there exists a proper normal
subgroup L of G such that G = KL and K ∩ L 6 KG = 1. Since
K ∩ N 6= 1 and K ∩ L = 1, we have N 6= L and thus N ∩ L = 1.
Consequently |NL| = |N ||L| = |K||L| = |KL| = |G| and thus
G = NL. Let M an maximal subgroup of G containing L , then
|G : M | = p since G/L is a p-group. Obviously G = NM and
N ∩ M = 1. But then |N | = |G : M | = p, a contradiction to (1).
(3) N is the unique minimal normal subgroup of G contained in E and
N � Φ(E).
Since dp > |N | by (2), it’s easy to verify that (G/N, E/N) still
satisfies the hypothesis of this theorem. The minimal choice of
(G, E) implies E/N is p-hypercyclically embedded in G/N . From
N. Su, Y. Wang 279
Lemma 2.10, N must be the unique minimal normal subgroup of G
contained in E. From Lemma 2.9, we have N � Φ(E).
(4) Final contradiction.
By (3), there exit a maximal subgroup M of E such that E = NM .
Ep = Ep ∩ NM = N(Ep ∩ M). Clearly Ep ∩ M < Ep since N is
not contained in M , so we can choose a maximal subgroup K of
Ep such that Ep ∩ M 6 K. Note that now Ep = NK, if N ∩ K = 1,
then by simple calculation we know that |N | = p, contradict to
(1). Hence 1 < N ∩ K < N . Clearly |N | < dp 6 |K|, so we can
choose a subgroup H with order dp such that 1 < N ∩ K < H 6 K.
Because N 6= H and N is the unique minimal normal subgroup
of G contained in E, we have HG = 1. By the hypothesis of this
Theorem, H is c-normal in E and hence there exit a normal subgroup
L of G such that G = HL and H ∩ L 6 HG = 1. Therefore
E = E ∩ HL = H(E ∩ L) and E ∩ L is a non trivial normal
subgroup of G contained in E. But since H ∩ (E ∩ L) 6 H ∩ L = 1
and H ∩ N 6= 1, we have N � E ∩ L, contradicts to N being the
unique minimal normal subgroup of G contained in E.
Remark. The conclusion of Theorem A does not hold if we replace
“c-normal” with “c-supplemented” in the hypothesis. One can take A5 for
a example. Obviously every subgroup of A5 with order 5 is c-supplemented
in A5, but A5 is not 5-hypercyclically embedded in itself.
Corollary 2.12. Let dp(G) be a prime power divisor of |Gp| satisfying
the following properties: If |Gp| = p then dp(G) = |Gp| = p; if |Gp| > p
then 1 < dp(G) < |Gp|. Suppose that all of the subgroups of Gp with order
dp(G) and 2dp(G) (if p = 2 and Gp is not an abelian nor a quaternion
free 2-group) are c-normal in G. Then G is p-supersolvable.
Corollary 2.13. Let E be a normal finite group of G and suppose that
G/E is p-supersolvable. Suppose that all of the subgroups of Ep with order
dp and 2dp (if p = 2 and Ep is not an abelian nor quaternion free 2-group)
are c-normal in G. Then G is p-supersolvable.
It is clear that G is p-nilpotent implies G is p-supersolvable but the
converse is not true. However, The following lemma reveals a connection
between p-nilpotent and p-supersolvable through the p-nilpotency of
NG(Gp).
Lemma 2.14. G is p-nilpotent if and only if G is p-supersolvable and
NG(Gp) is p-nilpotent.
280 c-normal subgroups
Proof. Suppose this lemma is not true and let G be a minimal counterex-
ample. Since NG/O
p′ (G)(GpOp′(G)/Op′(G)) = NG(Gp)Op′(G)/Op′(G), we
have that Op′(G) = 1 by induction.
Let N be a minimal normal subgroup of G. Then |N | = p since G is
p-supersolvable and Op′(G) = 1. It’s easy to verify that G/N still satisfy
the hypothesis of this lemma. Again from induction we know that N is the
unique minimal normal subgroup of G, Φ(G) = 1 and N = CG(N). But
the fact |N | = p implies that Gp 6 CG(N). Therefore we have Gp = N .
It follows that G = NG(Gp) is p-nilpotent, a contradiction.
Now we can prove Theorem B by using Theorem A and Lemma 2.14.
Proof of Theorem B. Suppose this is not true. Let (G, E) be a coun-
terexample such that |G| + |E| is minimal. We first claim that E is
p-nilpotent. Since NG(Ep) is p-nilpotent, NE(Ep) = NG(Ep) ∩ E is also
p-nilpotent. If Ep is abelian, then NE(Ep) = CE(Ep) and hence E is p-
nilpotent by Burnside’s theorem. If Ep is not abelian, then every subgroup
of Ep with order dp (dp is a prime power divisor of Ep and 1 < dp < |Ep|)
and 2dp (if p = 2 and Ep is not quaternion free) is c-normal in E by
hypothesis and Lemma 1.2(3). We know from Corollary 2.12 that E is
p-supersolvable. It follows from Lemma 2.14 that E is p-nilpotent.
By induction, we have Op′(E) = 1 and thus E must be a p-group.
Therefore G = NG(E) = NG(Ep) is p-nilpotent, a contradiction.
Remark. In Theorem A we ask Ep to be c-normal in G provided that
|Ep| = p. But we don’t impose the c-normality on Ep in Theorem B under
the same circumstance because Ep is abelian if |Ep| = p.
Corollary 2.15. Suppose NG(Gp) is p-nilpotent. If either Gp is abelian
or every subgroup of Gp with order dp (dp is a prime power divisor of Gp
and 1 < dp < |Gp|) and 2dp (if p = 2 and Gp is not quaternion free) is
c-normal in G, then G is p-nilpotent.
3. Applications
In this section, we give some applications to show that we can apply
our results to generalize some known results.
Corollary 3.1 ([1, Theorem 3.4]). Let F be a saturated formation con-
taining U . If all minimal subgroups and all cyclic subgroups with order 4
of GF are c-normal in G, then G ∈ F .
N. Su, Y. Wang 281
Proof. From Theorem A, we know that GF is p-hypercentrally embedded
in G for all p ∈ π(GF ) and thus GF 6 ZU(G). Since F be a saturated
formation containing U , we have that ZU(G) 6 ZF (G). Consequently
G ∈ F because G/GF ∈ F and GF 6 ZU (G) 6 ZF (G).
Corollary 3.2 ([8, Theorem 0.1]). Let E be a normal subgroup of a group
G of odd order such that G/E is supersolvable. Suppose that every non-
cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P |
and all subgroups H of P with order |H| = |D| are c-normal in G. Then
G is supersolvable.
Proof. Let p be the minimal prime divisor of |E|. If Ep is cyclic, then E
is p-nilpotent by [13, Lemma 2.8]. If Ep is not cyclic, then by Corollary
B, E is p-supersolvable and thus p-nilpotent since now p is the minimal
prime divisor of |E|. By repeating this argument we know that E has a
Sylow-tower and therefore E is solvable. Let p be any prime divisor of
|E|, If Ep is cyclic, then E is p-hypercentrally embedded in G since now
E is p-solvable. If Ep is not cyclic, E is also p-hypercentrally embedded
in G by Theorem A. As a result we have E 6 ZU (G). It follows that G is
supersolvable since G/E is supersolvable and E 6 ZU (G).
Corollary 3.3 ([5, Theorem 3.1]). Let p be an odd prime dividing the
order of a group G and P a Sylow-subgroup of G. If NG(P ) is p-nilpotent
and every maximal subgroup of P is c-normal in G, then G is p-nilpotent.
By noting the fact that if p is a prime such that (|G|, p − 1) = 1, then
G is p-nilpotent if and only if G is p-supersolvable, we have the following
two corollary:
Corollary 3.4 ([5, Theorem 3.4]). Let p be the smallest prime number
dividing the order of a group G and P a Sylow p-subgroup of G. If every
maximal subgroup of P is c-normal in G, then G is p-nilpotent.
Proof. If |P | = p, then G is p-nilpotent by [13, Lemma 2.8]. If |P | > p,
then by Corollary 2.12, G is p-supersolvable. Hence G is p-nilpotent.
Corollary 3.5 ([5, Theorem 3.6]). Let p be the smallest prime number
dividing the order of group G and P a Sylow p-subgroup of G. If every
minimal subgroup of P ∩ G′ is c-normal in G and when p = 2, either
every cyclic subgroup of P ∩ G′ with order 4 is also c-normal in or P is
quaternion-free, then G is p-nilpotent.
282 c-normal subgroups
Corollary 3.6 ([5, Corollary 3.9]). Let p be an odd prime number dividing
the order of a group G and P a Sylow p-subgroup of G. If every minimal
subgroup of P ∩ G′ is c-normal in G, then G is p-supersolvable.
References
[1] Ballester-Bolinches, A.; Wang, Yanming, Finite groups with some C-normal mini-
mal subgroups. J. Pure Appl. Algebra 153 (2000), no. 2, 121-127.
[2] Asaad, M.; Ramadan, M., Finite groups whose minimal subgroups are c-
supplemented. Comm. Algebra 36 (2008), no. 3, 1034-1040.
[3] Doerk,R. and Hawkes, T., Finite soluble groups, Walter De Gruyter, Berlin-New
York, 1992.
[4] Terence M. Gagen, Topics in finite groups, Cambridge University press, London,
1976.
[5] Guo, Xiuyun; Shum, K. P., On c-normal maximal and minimal subgroups of Sylow
p-subgroups of finite groups. Arch. Math. (Basel) 80 (2003), no. 6, 561-569.
[6] Dornhoff, Larry, M-groups and 2-groups. Math. Z. 100 1967 226-256.
[7] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994.
[8] Skiba, Alexander N., A note on c-normal subgroups of finite groups. Algebra
Discrete Math. 2005, no. 3, 85-95.
[9] Jaraden, Jehad J.; Skiba, Alexander N., On c-normal subgroups of finite groups.
Comm. Algebra 35 (2007), no. 11, 3776-3788
[10] Ballester-Bolinches, A.; Wang, Yanming; Xiuyun, Guo, c-supplemented subgroups
of finite groups. Glasg. Math. J. 42 (2000), no. 3, 383-389.
[11] Wang, Yanming, c-normality of groups and its properties. J. Algebra 180 (1996),
no. 3, 954-965.
[12] Weinstein, M., etl.(editor), Between nilpotent and solvable. Polygonal Publishing
House, Passaic (1982).
[13] Wei, Huaquan; Wang, Yanming, On c*-normality and its properties. J. Group
Theory 10 (2007), no. 2, 211-223.
Contact information
N. Su School of Mathematics, Sun Yatsen University,
Guangzhou, 510275, China
E-Mail(s): mc04sn@mail2.sysu.edu.cn
Y. Wang Lingnan College and School of Mathematics, Sun
Yatsen University, Guangzhou, 510275, China
E-Mail(s): stswym@mail.sysu.edu.cn
Received by the editors: 08.02.2013
and in final form 22.04.2013.
|
| id | nasplib_isofts_kiev_ua-123456789-154251 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-29T10:50:05Z |
| publishDate | 2015 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Ning Su Yanming Wang 2019-06-15T11:44:50Z 2019-06-15T11:44:50Z 2015 On c-normal and hypercentrally embeded subgroups of finite groups / Ning Su, Yanming Wang // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 270–282. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC:20D10. https://nasplib.isofts.kiev.ua/handle/123456789/154251 In this article, we investigate the structure of a finite group G under the assumption that some subgroups of G are c-normal in G The research has been supported by NSF China (11171353). en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On c-normal and hypercentrally embeded subgroups of finite groups Article published earlier |
| spellingShingle | On c-normal and hypercentrally embeded subgroups of finite groups Ning Su Yanming Wang |
| title | On c-normal and hypercentrally embeded subgroups of finite groups |
| title_full | On c-normal and hypercentrally embeded subgroups of finite groups |
| title_fullStr | On c-normal and hypercentrally embeded subgroups of finite groups |
| title_full_unstemmed | On c-normal and hypercentrally embeded subgroups of finite groups |
| title_short | On c-normal and hypercentrally embeded subgroups of finite groups |
| title_sort | on c-normal and hypercentrally embeded subgroups of finite groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154251 |
| work_keys_str_mv | AT ningsu oncnormalandhypercentrallyembededsubgroupsoffinitegroups AT yanmingwang oncnormalandhypercentrallyembededsubgroupsoffinitegroups |