c Journal “Algebra and Discrete Mathematics”
Finite groups with a system of generalized
Let H be a normal subgroup of a finite group G.
A number of authors have investigated the structure of G underthe assumption that all minimal or maximal subgroups in Sylowsubgroups of H are well-situated in G. A general approach to theresults of that kind is proposed in this article. The author has foundthe conditions for p-elements of H under which G-chief p-factors ofH are F-central in G.
All groups considered in this article will be finite. A number of authorshave investigated the structure of a non-nilpotent group G under the as-sumption that all minimal or maximal subgroups in Sylow subgroups ofG are well-situated in G. The first result in this direction was obtained byIto [1]; he proved that a group G of odd order is nilpotent provided that allminimal subgroups of G lie in the center of G. This result was developedby Gasch¨
utz in the following way: if every minimal subgroup of G is nor-
mal in G, then a Sylow 2-subgroup P of G′ is normal and G′/P is nilpo-tent (see [2], Theorem IV.5.7). Buckley [3] also considered the situationwhen minimal subgroups are normal; this means that these subgroupsare U-central normal subgroups where U is the formation of supersolublegroups. Later, some authors [4], [5], [6], [7] extended the mentioned re-sults using formation theory; they investigated groups in which minimalsubgroups lie in F-hypercenter of the group. Other generalizations were
2000 Mathematics Subject Classification: 20D10. Key words and phrases: finite group, Qf -central element, formation.
obtained in [8], [9] using the concept of a c-normal subgroup introducedin [10]. A subgroup H of G is called c-normal if there exists a normalsubgroup N of G such that G = HN and H ∩ N ⊆ HG = CoreG(H). Itis clear that if H = a is a c-normal primary cyclic subgroup of G, thenH/HG is either normal or normally complemented; in this case aB lies ina cyclic chief factor A/B of G. A more general approach was proposed ina paper [11] in which a concept of a QF-central element was introduced. An element a of a group G is called QF-central if there exists a F-centralchief factor A/B of G such that a ∈ A\B. Thus, the general line is toinvestigate a group with a system of QF-central elements. It is inter-esting that groups with given systems of complemented, supplementedor c-supplemented [12] minimal subgroups actually appear groups with asystem of QF-central elements. We recall that Gorchakov [13] proved thata group is supersoluble if all its minimal subgroups are complemented. We also mention articles [14], [15], [16], [17] in which the groups witha given system of complemented and S-quasinormal minimal subgroupsare investigated.
Analyzing the mentioned papers we can draw a conclusion that they
are connected with the solution of the following question.
Question A. Let F = LF (F ) be a saturated formation, H a normalsubgroup of a group G, p a prime. Assume that all elements of order pin H are QF-central in G. Assume also that if p = 2, then all elementsof order 4 in H are QF-central in G. Is it true that G/CG(A/B) ∈ F (p)for every G-chief factor A/B of H whose order |A/B| is divided by p?
Another line of investigations is concerned with maximal subgroups of
Sylow subgroups. So, Srinivasan [18] proved that a group G is supersolu-ble if maximal subgroups of its Sylow subgroups are normal in G. Clearly,under assumptions of Srinivasan’s theorem every Sylow subgroup P of Gsatisfies the following condition: every element in P \Φ(P ) is QU-centralin G. Srinivasan’s theorem was generalized in [8], [10], [19] by replacingthe normality with the weaker condition of c-normality; besides, in [8] thecondition of c-normality of maximal subgroups of Sylow subgroups in anormal nilpotent subgroup is considered. We also recall S.N.Chernilov’sresult [20] on supersolubility of a group G with abelian Sylow subgroupshaving the following property: every primary cyclic subgroup comple-mented in a Sylow subgroup of G is complemented in G. Vedernikovand Kuleshov [21] established that a group G is supersoluble if every itsprimary cyclic subgroup having a non-trivial supplement in a Sylow p-subgroup of G possesses a non-trivial supplement in G. Analyzing thisline of investigations we arrive at the conclusion that they are concernedwith the following question.
Question B. Let F be a saturated formation, H a normal subgroup of agroup G. Assume that every Sylow subgroup P of H satisfies the followingcondition: every element in P \(Φ(P ) ∪ Φ(G)) is QF-central in G. Is ittrue that every non-Frattini G-chief factor of H is F-central in G?
The main aim of the present article is to give a positive answer to
Questions A and B. Moreover, we give the answer even in a more generalform fixing our attention to the behaviour of p-elements with a fixedprime p.
We use the standard notations [22], [23]. For a prime p, Gp denotes aSylow p-subgroup of G; π(G) is the set of primes dividing |G|; π(F) =
π(G); F ∗(G) is the generalized Fitting subgroup of G, i.e. the
quasinilpotent radical of G [24]. A subgroup M is called a minimal sup-plement to a normal subgroup H of G if M H = G and M1H = G forevery proper subgroup M1 of M.
We need some information from the theory of formations. A for-
mation is a class of groups closed under taking homomorphic imagesand subdirect products. We denote by GF a F-residual of a group G,i.e. the smallest normal subgroup with quotient in F. A formation Fis called saturated if G/Φ(G) ∈ F always implies G ∈ F. A functionf : {primes} → {f ormations} is called a local satellite. A chief factorH/K of G is called f -central in G if G/CG(H/K) ∈ f(p) for every primep dividing |H/K|.
If F is the class of all groups whose chief factors are f -central, then
we say that f is a local satellite of F and write F = LF (f ). A localsatellite f of a formation F is called: 1) full if f (p) = Npf(p) for everyprime p (here Np is the class of p-groups); 2) integrated if f(p) ⊆ F forevery prime p; 3) semi-integrated if for every prime p, f (p) is either asubformation in F or the class E of all groups; 4) canonical if it is fulland integrated; 5) semicanonical if it is full and semi-integrated. Thenotation LF (F ) means that F is a canonical local satellite of LF (F ). Achief factor H/K of G is called F-central in G if it is f -central, where fis the canonical local satellite of F.
In the proofs of our results we will use the following theorems.
2.1. Every non-empty saturated formation possesses a semicanonical localsatellite and the unique canonical local satellite (see [22], [23], [25]).
2.2. Let f be a local satellite such that f (p) = (1) and f (q) = E for everyprime q = p. Then LF (f ) is the class of p-nilpotent groups [2].
2.3.(a) Let f be a satellite such that f (p) is the class of all abelian groupswith exponents dividing p − 1, and f (p) = E for every prime q = p. ThenLF (f ) is the class p-U of p-supersoluble groups.
(b) Let a prime p divide the order of a chief factor H/K of G. Then
H/K is p-U-central if and only if |H/K| = p ( [2], Kapitel VI).
2.4. Let F be a saturated formation and H a normal subgroup of a groupG such that H/H ∩Φ(G) ∈ F. Then H = A×B where A ∈ F, B ⊆ Φ(G),π(B) ∩ π(F) = ∅ ( [23], Theorem 4.2).
2.5. Let H be a normal subgroup of G such that H/H ∩ Φ(G) is p-nilpotent. Then H is p-nilpotent [2], [23].
2.6. Let F = LF (f ) where f is semi-integrated. Let H/L be a G-chieffactor of GF such that f (p) ⊆ F for some p ∈ π(H/L). Then H/L isf -eccentric in G if one of the following conditions holds: 1) H/L is non-Frattini in G; 2) a Sylow p-subgroup in GF is abelian ( [23], Theorems8.1 and 11.6).
2.7. If G = AB, then for every prime p there exist Sylow p-subgroupsAp, Bp and Gp in A, B and G such that Gp = ApBp ( [23], p. 134).
2.8. Let H be a normal subgroup of a group G. Let α and β be G-chiefseries of H. Then there exists a one-to-one correspondence between thechief factors of α and those of β such that the corresponding factors areG-isomorphic and such that the Frattini chief factors of α correspond tothe Frattini chief factors of β ( [22], Theorem A.9.13).
2.9. Let H be a normal subgroup of a group G, and let M be a minimalsupplement to H in G. If M contains at least one Sylow p-subgroup ofH for some prime p, then H is p-nilpotent ( [26]; [23], Theorem 12.4).
2.10. Let F be a saturated formation, and H a normal subgroup of agroup G such that every G-chief factor of H is F-central in G. ThenG/CG(H) ∈ F (see [23], Theorem 9.5).
2.11. If G is a group, then CG(F ∗(G)) ⊆ F (G) (see [25], Theorem15.22).
2.12. (a) Let p be an odd prime. A group G is p-nilpotent if every itselement of order p is Q-central in G.
(b) A group G is 2-nilpotent if every its 2-element of order ≤ 4 is
Q-central in G (see [11], Theorem 2).
2.13. Let F be a saturated formation and H a normal subgroup of a groupG. Let ω be the set of primes p such that HF possesses an abelian Sylowp-subgroup. Then there exists a subgroup C of G such that G = CHF andπ(C ∩ HF) ∩ ω = ∅ (see [23], Theorem 11.8).
If a Sylow p-subgroup of a p-soluble group G is abelian, then
lp(G) ≤ 1 (see [23], Theorem 5.11).
2.15. Let G = a B, where a = 1 is a 2-subgroup and B = G. Thenthere exists a normal maximal subgroup M of G such that G = a M(see [21], Lemma 1).
a p-subgroup. Assume that Hp is abelian and G is p-soluble. Then a is aQU-central element of G.
Proof. Let G be a counterexample of minimal order. Then we can assumethat BG = Op′(H) = 1. By 2.14, Hp is normal in G. We have that
Evidently, Hp∩B is normal in G. Since BG = 1, it follows that Hp∩B = 1,and Hp = a is normal in G.
For a prime p and a group H, we use the following conventions:
Wp(H) = {x : x ∈ H, |x| = p} if p is odd,W2(H) = {x : x ∈ H, |x| ∈ {2, 4}},W (H) = {x : x ∈ H, |x| is a prime or |x| = 4}.
Definition 3.1 (see [11], Definition 3). Let f be a local satellite. Anelement a of a group G is called Qf -central in G if there exists a f -central chief factor A/B of G such that a ∈ A\B. The identity elementis always regarded as a Qf -central element.
Definition 3.2. Let F = LF (f ) be a saturated formation, where f is anintegrated local satellite of F. An element a of G is called QF-central ifit is Qf -central.
It is easy to show that Definition 3.2 does not depend on the choice
Definition 3.3. An element a of G is called Q-central if it is QN-central(this means that there exists a central chief factor A/B of G such thata ∈ A\B).
Theorem 3.1. Let p be a prime, and F = LF (f ) a saturated formation,where f is a semicanonical local satellite such that f (p) ⊆ F and f (q) = Efor every prime q = p. Let H be a normal subgroup of a group G. Assumethat every element in Wp(H) is Qf-central in G. Then every G-chieffactor of H is f -central in G.
Proof. We will use induction on |G|+|H|. Let W = Wp(H) = {xi : i ∈ I}. We may assume that W = ∅. Assume that there is a normal p′-subgroupK = 1 in G. Consider the natural epimorphism α : G → G/K. Evidently,W α = Wp(HK/K). If xi ∈ W , then by assumption,there is a f-centralchief factor A/B of G such that xi ∈ A\B. The factors AK/BK andA/B(A ∩ K) are G-isomorphic; besides, it follows from xi ∈ A\B thatA = B(A ∩ K) because every p-element in B(A ∩ K) is contained in B. Hence, B = B(A ∩ K). We have that xi ∈ AK\BK, and AK/BK is af -central chief factor of G. But then, (AK)α/(BK)α is a f -central chieffactor of Gα. Clearly, xαi ∈ (AK)α\(BK)α. By the inductive hypothesis,the theorem is true for G/K. Then it is also true for G. So, we mayassume that Op′(G) = 1.
Consider an arbitrary element xi in W . By assumption, there is a
f -central chief factor A/B of G such that xi ∈ A\B. We have
AH/BH ≃ A/A ∩ BH = A/B(A ∩ H).
Since xi ∈ A\B, the equality B = B(A ∩ H) is impossible. Therefore,
A = B(A∩H), and we have that A/B and A∩H/B ∩H are G-isomorphicG-chief factors. So, we showed that for each xi ∈ W , there is a G-chieffactor Xi/Yi of H such that xi ∈ Xi\Yi and Xi/Yi is f-central in G. Set
Clearly, G/C ∈ f (p) ∈ F. Therefore, every G-chief factor of HC/C is
f -central. If H ∩ C = H, then, by the inductive hypothesis, every G-chieffactor of H ∩ C is f -central in G, and the theorem is proved. So, we mayassume that H ⊆ C. This means that every element in W is Q-centralin C. By 2.12, H is p-nilpotent. Since Op′(G) = 1, we have that H isa p-group. Let Q be a Sylow q-subgroup in C, q = p. Then we havethat every element in W is Q-central in QH. By 2.12, QH is p-nilpotent. Since G/C ∈ f (p) = Npf(p) and Cq centralizes every G-chief p-factor ofH for every prime q = p, it follows that every G-chief p-factor of H isf -central in G. The theorem is proved.
Corollary 3.1.1. Let F be a saturated formation, and G a group suchthat every element in W (G) is QF-central in G. Then G ∈ F.
Proof. Applying Theorem 1 for the case H = G and for the arbitraryprime p, we obtain that every chief factor of G is F-central. So, G ∈ F,and the result is true.
Corollary 3.1.2. Let F be a saturated formation, and H a normal sub-group of a group G such that G/H ∈ F and every element in W (F ∗(H))
Proof. By Theorem 3.1, every G-chief factor of F ∗(H) is F-central in G. By 2.10, G/CG(F ∗(H)) belongs to F. From this and from G/H ∈ F itfollows that G/CH(F ∗(H)) ∈ F. By 2.11, CH(F ∗(H)) is contained inF ∗(H). Therefore G/F ∗(H) belongs to F, and we have that G ∈ F.
Corollary 3.1.3. Let p be a prime, and H a normal subgroup of a groupG. If every element in Wp(H) is Q-central in G, then H is p-nilpotent,and H/Op′(H) lies in the hypercenter of G/Op′(H).
Corollary 3.1.4. Let p be a prime, and H a normal subgroup of a groupG. Assume that every element in Wp(H) is QU-central in G. Then H isp-supersoluble, and every G-chief p-factor of H is cyclic.
We introduce the subgroup Op′,Φ(G) as follows:
Op′,Φ(G)/Op′(G) = Φ(G/Op′(G)).
Theorem 3.2. Let p be a prime, and F = LF (f ) a saturated formation,where f is a semicanonical local satellite such that f (p) ⊆ F and f (q) = Efor every prime q = p. Let H be a normal subgroup of a group G. Assumethat every element in Hp\(Φ(Hp)∪Φ(G)) is Qf-central in G. Then everyG-chief factor of H/H ∩ Op′,Φ(G) is f-central in G/H ∩ Op′,Φ(G).
Proof. We will prove this theorem using induction on |H| + |G|. Assumethat there is a normal p′-subgroup K = 1 in G such that K ⊆ H. Consider the natural epimorphism α : G → G/K. If a ∈ Hp, thenaα belongs to a Sylow p-subgroup Hα
p ) ∪ Φ(Gα). Since HpK/K ≃ Hp, it follows that a is
not contained in Φ(Hp). Furthermore, it follows from (Φ(G))α ⊆ Φ(Gα)that a is not contained in Φ(G). By assumption, there is a f -centralchief factor A/B of G such that a ∈ A\B. The factors AK/BK andA/B(A ∩ K) are G-isomorphic; besides, it follows from a ∈ A\B thatA = B(A ∩ K) because every p-element in B(A ∩ K) is contained in B. Hence, B = B(A ∩ K). We have that a ∈ AK\BK, and AK/BK is af -central chief factor of G. But then, (AK)α/(BK)α is a f -central chieffactor of Gα. Clearly, aα ∈ (AK)α\(BK)α. By the inductive hypothesis,the theorem is true for G/K. Then it is also true for G.
So, we may assume that Op′(H) = 1. Consider H ∩ GF. We may
assume that H ∩ GF has non-Frattini G-chief factors. We call a normalsubgroup L of G f -limit if L/L∩Φ(G) is a f -eccentric G-chief factor. Theset Σ of f -limit subgroups contained in H ∩ GF is not empty. Really, ifL/(Φ(G) ∩ H ∩ GF) is a minimal normal subgroup in G/(Φ(G) ∩ H ∩ GF),where L ⊆ H ∩ GF, then L is f -limit by 2.6. So, let L be a subgroup
of minimal order in Σ. Set Φ = L ∩ Φ(G). It follows from Op′(H) = 1and 2.5 that p divides |L/Φ|. Let M/Φ be a minimal supplement forL/Φ in G/Φ. Then by 2.7 we have Gp = MpLp. By 2.9, Mp/Φ doesnot contain Lp/Φ; hence, M = G. From Gp = MpLp it follows thatHp = Gp ∩ H = (Hp ∩ Mp)Lp, where Hp ∩ Mp does not contain Lp. Hence, there is an element a in Lp\(Hp ∩ Mp) such that a ∈ Φ(Hp). Since Hp ∩ Mp ⊇ Φ = L ∩ Φ(G), we have that a ∈ Φ(G). So, we get
By assumption, G possesses a f -central chief factor A/B such that
a ∈ A\B. Consider AL/BL ≃ A/B(A ∩ L). Since A/B is a chief factor,B(A ∩ L) is either equal B or else A. Since a belongs to A ∩ L and doesnot belong to B, we have that B = B(A ∩ L). Hence, A = B(A ∩ L). So, we have G-isomorphic G-chief factors A/B and A ∩ L/B ∩ L; besides,a ∈ (A ∩ L)\(B ∩ L).
Suppose that A∩L/B ∩L is a non-Frattini chief factor of G. Then, by
2.8, A∩L/B ∩L is G-isomorphic with L/Φ. In this case, L/Φ is f -centralin G. This contradicts 2.6. So, we obtained that A ∩ L/B ∩ L is a Frattinichief factor of G. If B ∩ L is not contained in Φ(G), we have that B ∩ Lpossesses a f -limit normal subgroup of G; this contradicts the minimalityof |L|. Therefore, B ∩ L ⊆ Φ(G). We get A ∩ L ⊆ Φ(G). Hence,a ∈ A ∩ L ⊆ Φ(G). We arrive at a contradiction, because a ∈ Φ(G). Thetheorem is proved.
Corollary 3.2.1. Let F be a saturated formation, H a normal subgroupof a group G such that G/H ∈ F and for every prime p the followingcondition holds: each element in Hp\(Φ(Hp) ∪ Φ(G)) is QF-central in G. Then G ∈ F.
Corollary 3.2.2. Let F be a saturated formation, H a normal solublesubgroup of a group G such that G/H ∈ F and the following conditionholds: if P is a Sylow subgroup of F (H), then each element in P \(Φ(P )∪Φ(G)) is QF-central in G. Then G ∈ F.
Proof. Let Φ = Φ(G) ∩ F (H). By Theorem 3.2, every G-chief fac-tor of F (H)/Φ is F-central in G. By 2.5, F (H)/Φ = F (H/Φ). By2.10, G/Φ/CG/Φ(F (H/Φ)) ∈ F. Therefore, G/CG(F (H)/Φ) ∈ F. Fromthis and from G/H ∈ F it follows that G/CH(F (H)/Φ) ∈ F. ButCH(F (H)/Φ) ⊆ F (H). Hence, G/Φ ∈ F. Since F is saturated, wehave that G ∈ F.
Corollary 3.2.3. Let p be a prime, and H be a normal subgroup of agroup G. Assume that every element in Hp\(Φ(Hp) ∪ Φ(G)) is Q-central
in G. Then H is p-nilpotent, and every its non-Frattini G-chief p-factoris central in G.
Corollary 3.2.4. Let p be a prime, and H be a normal subgroup of agroup G. Assume that every element in Hp\(Φ(Hp)∪Φ(G)) is QU-centralin G. Then H is p-supersoluble, and every its non-Frattini G-chief p-factor is cyclic.
Corollary 3.2.5. A group G is supersoluble if for every non-cyclic Sy-low subgroup P of G the following condition holds: every element inP \(Φ(P ) ∪ Φ(G)) is QU-central in G.
Proof. If G2 is non-cyclic, then by Theorem 3.2, G is 2-supersoluble. IfG2 is cyclic, then G is 2-nilpotent. Thus, G is soluble, and by Theorem3.2, G is p-supersoluble for all prime p such that Gp is non-cyclic.
Definition 3.4. An element a = 1 of an abelian group P is called basicif there exists a subgroup B in P such that P = a × B. We denote byB(P ) the set of all basic elements in P.
The following theorem generalizes S.N.Chernikov’s result [20] on a
finite group with a system of complemented subgroups.
Theorem 3.3. Let p be a prime, and F = LF (f ) a saturated formationof p-soluble groups, where f is a semicanonical local satellite such thatf (p) ⊆ F and f (q) = E for every prime q = p. Let H be a normalsubgroup of a group G. Assume that Hp is abelian and every element inB(Hp) is Qf-central in G. Then every G-chief factor of H is f-centralin G.
Proof. We will use induction on |H| + |G|. As well as in the proof
of Theorem 3.2, it is easy to show that the assumption of the theorem isvalid for G/Op′(H) and H/Op′(H). So, we may assume that Op′(H) = 1. We will consider two cases: H = G and H = G.
Case 1. Assume that H = G. Consider the F-residual R of G. By
2.13, there exists a subgroup C such that G = CR and p does not divide|C ∩ R|. By 2.7, Gp = CpRp and Cp ∩ Rp ⊆ C ∩ R. So, Gp = Cp × Rp. It follows from this that B(Rp) ⊆ B(Gp).
It is clear that B(Rp)\Φ(G) = ∅. Consider a ∈ B(Rp)\Φ(G). By
assumption, there is a f -central chief factor A/B of G such that a ∈A\B. We have that AR/BR is G-isomorphic with A/B(A ∩ R). Sincea ∈ A\B, it follows that A = B(A ∩ R). Thus, A/B and A ∩ R/B ∩ Rare G-isomorphic f -central G-chief factors. This contradicts 2.6. So, thetheorem is true for the case H = G.
Case 2. Now we assume that H = G. Let H be the formation of p-
soluble groups. By 2.13, there exists a subgroup C such that G = CHH
and p does not divide |C ∩ HH|. By 2.7, Gp = CpHH
p are Sylow p-subgroups in G, C and H H. Evidently, Gp ∩ H = Hp is
a Sylow p-subgroup of H. Furthermore, Gp ∩ HH = HH
tion of the theorem there exists a f -central chief factor A/B of G suchthat a ∈ A\B. Since all groups in f (p) are p-soluble, A/B is a p-group. Consider
AHH/BHH ≃ A/A ∩ BHH = A/B(A ∩ HH).
Since a ∈ (A ∩ HH)\B, we have that B = B(A ∩ HH). Therefore,A = B(A∩HH). We have that A/B and A∩HH/B∩HH are G-isomorphicG-chief factors. Since Hp is contained in CH(A ∩ HH/B ∩ HH) we havethat
Thus, there is a H-central chief p-factor D1/D2 of H such that
A ∩ HH ⊇ D1 ⊃ D2 ⊇ B ∩ HH.
p ) is empty and H is p-soluble. Since Op′ (H ) =
1 and Hp is abelian, it follows by 2.14 that Hp is normal. Clearly, we canassume that H is a p-group. Let B(H) = {xi : i ∈ I}. By assumption,for every i ∈ I there exists a f -central G-chief factor Ai/Bi such thatxi ∈ Ai\Bi. We set
Then factors AiH/BiH and Ai/BiXi are G-isomorphic. Since xi ∈ Xi\B,we have that Bi = BiXi. Thus, Ai = BiXi. So, Ai/Bi and Xi/Yiare G-isomorphic f -central chief factors of G. It follows from this thatG/CG(Xi/Yi) ∈ f(p). We have that
It follows from this that every element xi in B(H) is Q-central in HCq
for every prime q = p. For HCq the theorem is true (we note that byCase 1, the theorem is true if a considered normal subgroup coincideswith the whole group). Applying the proved part of the theorem to HCqand the formation of p-nilpotent groups we have that HCq is p-nilpotent. Therefore, CC(X/Y ) is a p-group for every G-chief factor X/Y of H. ButG/C ∈ f (p). We see that G/CG(X/Y ) belongs to Npf(p) = f(p), thatis X/Y is f -central in G. The theorem is proved.
Corollary 3.3.1. Let p be a prime. Assume that a normal subgroup Hof a group G possesses an abelian Sylow p-subgroup P . Assume also thatevery element in B(P ) is QU-central in G. Then H is p-supersoluble, andevery G-chief p-factor of H is cyclic.
Corollary 3.3.2. Let p be a prime. Assume that a normal subgroup Hof a group G possesses an abelian Sylow p-subgroup P . Assume also thatevery element in B(P ) is Q-central in G. Then H is p-nilpotent, andevery G-chief p-factor of H is central in G.
Corollary 3.3.3. Let H be a normal subgroup of a group G. Assumethat a Sylow 2-subgroup P of H is abelian and has the following property:
a is complemented in G for every a ∈ B(P ). Then H is 2-nilpotent,
and every its G-chief 2-factor is central in G.
Proof. By 2.15, every element in B(P ) is Q-central in G. Now we applyCorollary 3.3.2.
Corollary 3.3.4. Let H be a normal subgroup of a group G. Assumethat for every Sylow subgroup P of G the following condition holds: P isabelian, and a is complemented in G for every a ∈ B(P ). Then H issupersoluble, and every its G-chief factor is cyclic.
Proof. By Corollary 3.3.3, H is 2-nilpotent. So, H is soluble. Let P bea Sylow p-subgroup of H, p ∈ π(H). By assumption, for every a ∈ B(P )we have that
By 2.16, a is QU-central in G. Now we apply Corollary 3.3.1.
Corollary 3.3.5 (see [20]). Assume that every Sylow p-subgroup P of Gis abelian and satisfies the following condition: if a ∈ B(P ), then a iscomplemented in G. Then G is supersoluble.
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Russian Economic Academy named afterG. V. Plekhanov, Stremyanny per.
113054 Moscow, RussiaE-Mail: [email protected]
Received by the editors: 12.04.2004and final form in 06.12.2004.

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