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Publicly Available Published by De Gruyter May 7, 2019

Finite groups in which every cyclic subgroup is self-normalizing in its subnormal closure

Guohua Qian EMAIL logo
From the journal Journal of Group Theory

Abstract

For a given prime p, a finite group G is said to be a π’ž~p-group if every cyclic p-subgroup of G is self-normalizing in its subnormal closure. In this paper, we get some descriptions of π’ž~p-groups, show that the class of π’ž~p-groups is a subgroup-closed formation and that Op′⁒(G) is a solvable p-nilpotent group for every π’ž~p-group G. We also prove that if a finite group G is a π’ž~p-group for all primes p, then every subgroup of G is self-normalizing in its subnormal closure.

1 Introduction

In this paper, G always denotes a finite group. For a subgroup H of G, we denote by SG⁒(H) the subnormal closure of H in G, that is, the unique minimal subnormal subgroup of G containing H.

Definition 1.1.

Let G be a finite group and let p be a given prime.

  1. Following [1], G is said to be a π’ž-group (respectively, a π’žp-group) if every subgroup (respectively, every p-subgroup) H of G is self-normalizing in SG⁒(H).

  2. G is said to be a π’ž~-group (respectively, a π’ž~p-group) if every cyclic subgroup of prime power order (respectively, every cyclic p-subgroup) of G is self-normalizing in its subnormal closure.

It is shown in [1] that a π’žp-group G is p-solvable with lp⁒(G), the p-length of G, at most 1. In particular, all π’ž-groups are solvable. Clearly, a π’ž-group is necessarily a π’žp group for every prime p. Naturally, one may ask if the converse is true. The answer is positive. Actually, we get the following result.

Theorem 1.2.

For a finite group G and a given prime p, the following equivalences hold:

  1. G is a π’žp-group if and only if G is a π’ž~p-group.

  2. G is a π’ž-group if and only if G is a π’ž~-group.

For a given prime p, it is shown in [1, Theorem 8] that G is a π’žp-group if and only if G is p-solvable with lp⁒(G)≀1, and COp⁒(SG⁒(H))⁒(H)=1 for all subgroups H of G. The following descriptions of π’žp-groups appear more clearly.

Theorem 1.3.

For a finite group G and a prime p, the following statements are equivalent:

  1. G is a π’žp-group.

  2. G is p-solvable with lp⁒(G)≀1, and if a p-subgroup P of G acts on a pβ€²-subgroup V of G, then V=CV⁒(P)⋉[V,P], that is, C[V,P]⁒(P)=1.

  3. Every subgroup H of G can be expressed as H=NH⁒(P)⋉B, where P is a Sylow p-subgroup of H and B is a normal pβ€²-subgroup of H.

In Theorem 1.3 (3), since P acts coprimely on B, we have ([3, Theorem 8.2.7])

CB⁒(P)≀NH⁒(P)∩B and B=[B,P]⁒CB⁒(P).

This implies CB⁒(P)=1 and B=[B,P].

Combining Theorems 1.2 and 1.3, we get some descriptions of π’ž-groups. As shown in [1, Theorem A], the class of π’ž-groups is a subgroup-closed formation. Actually, this is a direct corollary of Theorem 1.2 and Theorem 1.4.

Theorem 1.4.

For a given prime p, the class of Cp-groups is a subgroup-closed formation.

Let 𝐅i⁒(G) be the ith ascending Fitting subgroup of G, that is, 𝐅0⁒(G)=1, 𝐅1⁒(G)=𝐅⁒(G) and 𝐅i+1⁒(G)/𝐅i⁒(G)=𝐅⁒(G/𝐅i⁒(G)). The Fitting height of a solvable group G is the smallest integer k such that 𝐅k⁒(G)=G. For a given prime p, we denote by Ξ©p⁒(G) the characteristic subgroup of G generated by all elements of order p. Clearly, Ξ©p⁒(G)≀Op′⁒(G).

Theorem 1.5.

For a Cp-group G, the following statements hold:

  1. Op′⁒(G) is solvable and p-nilpotent.

  2. Ωp⁒(G) has Fitting height at most 2.

Let G be a π’ž-group. By Theorem 1.5 (2), we get that Ω⁒(G), the characteristic subgroup of G generated by all elements of prime order, has Fitting height at most 2.

All unexplained notation is standard and taken mainly from [3]. The proofs given here are completely self-contained and independent of [1].

2 π’ž~p-groups and π’žp-groups

In this section, we will investigate π’ž~p-groups and π’žp-groups. For a subgroup D of G, it is easy to see that SG⁒(D)=SSG⁒(D)⁒(D), and this fact will be used freely.

Lemma 2.1.

For a C~p-group G, it follows that

  1. every subgroup of G is a π’ž~p-group,

  2. every quotient group of G is also a π’ž~p-group,

  3. G is p-solvable with lp⁒(G)≀1.

Proof.

(1) Let H be a subgroup of G, and let P be a cyclic p-subgroup of H. Since SG⁒(P) is subnormal in G, we have that SG⁒(P)∩H is subnormal in H. This implies SH⁒(P)≀SG⁒(P)∩H≀SG⁒(P). Now the self-normalization of P in SG⁒(P) yields NSH⁒(P)⁒(P)=P. Hence H is a π’ž~p-group.

(2) Let N⊴G, and let U/N be a cyclic p-subgroup of G/N. To see the required result, we may assume by induction that N is a minimal normal subgroup of G, and we only need to show the self-normalization of U/N in SG/N⁒(U/N). Assume that SG/N(U/N):=H/N<G/N. Since H is also a π’ž~p-group by (1), we get by induction that H/N is again a π’ž~p-group. Thus NH/N⁒(U/N)=U/N, as required. Therefore, we may assume that SG/N⁒(U/N)=G/N.

Let P be a cyclic p-subgroup of G such that U=P⁒N. Note that

NSG⁒(P)⁒(P)=P.

Suppose that SG⁒(P)=G. Then NG⁒(P)=P, so P∈Sylp⁒(G). Let us investigate NG⁒(P⁒N). Clearly, NG⁒(P)≀NG⁒(P⁒N), and thus NG⁒(P)=NNG⁒(P⁒N)⁒(P). By the Frattini argument and the normality of PN in NG⁒(P⁒N), we get

NG⁒(P⁒N)=P⁒N⁒NNG⁒(P⁒N)⁒(P)=P⁒N⁒NG⁒(P)=P⁒N.

Clearly, NG/N⁒(P⁒N/N)=NG⁒(P⁒N)/N. This yields the required result

NG/N⁒(U/N)=NG/N⁒(P⁒N/N)=NG⁒(P⁒N)/N=P⁒N/N=U/N.

Suppose that SG⁒(P)<G. Let M be a maximal normal subgroup of G such that SG⁒(P)≀M. Since U/N≀M⁒N/N⊴G/N and SG/N⁒(U/N)=G/N, we have G/N=M⁒N/N. Now the minimal normality of N implies G=MΓ—N. Note that M is a π’ž~p-group by (1), and so is G/N. It follows that U/N is self-normalizing in G/N, and we are done.

(3) Note that all subgroups and all quotient groups of G are also π’ž~p-groups. Suppose that G is not p-solvable. To see a contradiction, we may assume by induction that G is a nonabelian simple group of order divisible by p. Let P be a subgroup of G of order p. Since NG⁒(P)=P by the hypothesis, we have P∈Sylp⁒(G) and NG⁒(P)=CG⁒(P). It follows by [3, Theorem 7.2.1] that G has a normal p-complement, a contradiction. Consequently, G is p-solvable.

Suppose that lp⁒(G)β‰₯2. To get a contradiction, we may assume, by somewhat more standard inductive arguments, that the following occur:

  1. G has a unique minimal normal subgroup, say E;

  2. E=Op⁒(G) and Φ⁒(G)=1;

  3. G=M⋉E for a maximal subgroup M of G, and M=P⋉K, where M=Op′⁒(M), K=Op′⁒(K) and 1<P∈Sylp⁒(M) is cyclic. (For example, we show the cyclicity of P. Assume that P is not cyclic. Then G possesses different maximal normal subgroups, say A1,A2. Since lp⁒(A1)≀1 and lp⁒(A2)≀1 by induction, we have

    lp⁒(G)=lp⁒(A1⁒A2)=max⁑{lp⁒(A1),lp⁒(A2)}≀1,

    a contradiction.)

Now it is easy to see that G is the subnormal closure of P in G. Then we have NG⁒(P)=P by the hypothesis. However, P<P⋉E∈Sylp⁒(G), a contradiction. Thus lp⁒(G)≀1. ∎

Lemma 2.2.

Let D and V be subgroups of G with (|D|,|V|)=1. Assume that G=D⋉V and that D is self-normalizing in SG⁒(D). Then

C[V,D]⁒(D)=1,V=CV⁒(D)⋉[V,D],G=NG⁒(D)⋉[V,D]β€ƒπ‘Žπ‘›π‘‘β€ƒNG⁒(D)=DΓ—CV⁒(D).

Proof.

Since D acts coprimely on V, we have V=[V,D]⁒CV⁒(D) by [3, Theorem 8.2.7]. Clearly, [V,D] and D⋉[V,D] are normal in G. Note that (see [3, Theorem 8.2.7]) [V,D,D]=[V,D]. It follows that SG⁒(D)=D⋉[V,D].

Clearly, C[V,D]⁒(D)=N[V,D]⁒(D). Since NSG⁒(D)⁒(D)=D by the hypothesis, we get

CV⁒(D)∩[V,D]=C[V,D]⁒(D)=N[V,D]⁒(D)=1.

This implies V=[V,D]⁒CV⁒(D)=CV⁒(D)⋉[V,D]. Also, we get

NG⁒(D)=NG⁒(D)∩G=NG⁒(D)∩(D⁒CV⁒(D)⋉[V,D])=D⁒CV⁒(D)⁒(NG⁒(D)∩[V,D])=D⁒CV⁒(D)⁒N[V,D]⁒(D)=D⁒CV⁒(D)=DΓ—CV⁒(D),

and G=NG⁒(D)⋉[V,D]. ∎

Lemma 2.3.

For a given prime p, the group G is a Cp-group if and only if G is a C~p-group.

Proof.

We need only to show the β€œif” part. Let G be a π’ž~p-group, and let P be a p-subgroup of G. To see that P is self-normalizing in SG⁒(P), we may assume that P is not cyclic and that SG⁒(P)=G by induction. Notice that G is p-solvable with lp⁒(G)=1. Now SG⁒(P)=G implies

G=Op′⁒(G) and G=P⋉V

for a normal pβ€²-subgroup V of G. We only need to show that CV⁒(P)=1 since NV⁒(P)=CV⁒(P) and NG⁒(P)=P⁒NV⁒(P).

Let x∈CV⁒(P), and let D be a maximal subgroup of P. Clearly, both [V,D] and D⁒[V,D] are normal in G. Note that G/D⁒[V,D]=Op′⁒(G/D⁒[V,D]). Hence

G/D⁒[V,D]=SG/D⁒[V,D]⁒(P⁒[V,D]/D⁒[V,D]).

Since D>1 by the non-cyclicity of P, we have |G/D⁒[V,D]|<|G|. It follows by induction that G/D⁒[V,D] is a π’žp-group. Consequently, P⁒[V,D]/D⁒[V,D] is self-normalizing in G/D⁒[V,D]. This implies

x∈P⁒[V,D],
x∈P⁒[V,D]∩CV⁒(P)=[V,D]∩CV⁒(P)=C[V,D]⁒(P)≀C[V,D]⁒(D).

Observe that D⋉V is also a π’žp-group by induction; therefore, by Lemma 2.2, we get C[V,D]⁒(D)=1. Hence x=1, CV⁒(P)=1, and the proof is complete. ∎

Corollary 2.4.

For a finite group G and a given prime p, the following statements hold:

  1. Assume that G is a π’žp-group. Then all subgroups and all quotient groups of G are π’žp-groups; also, G is p-solvable with lp⁒(G)≀1.

  2. Assume that G is a π’ž~-group. Then all subgroups and all quotient groups of G are π’ž~-groups, G is solvable with lr⁒(G)≀1 for all primes r, and all subgroups of prime power order of G are self-normalizing in their subnormal closures.

Proof.

The results follow by Lemma 2.1 and Lemma 2.3. ∎

Proof of Theorem 1.3.

(1) β‡’ (2). Assume that G is a π’žp-group. By Corollary 2.4, G is p-solvable with lp⁒(G)≀1. Let P and V be a p-subgroup and a pβ€²-subgroup, respectively, of G such that P acts on V. Since P⋉V is again a π’žp-group by Corollary 2.4, the required result follows by Lemma 2.2.

(2) β‡’ (3). Let H≀G, and let P∈Sylp⁒(H). We may assume that P>1. Clearly, H is p-solvable with lp⁒(H)=1. Then Op′⁒(H)=P⋉B for a normal pβ€²-subgroup B of H. Clearly, [B,P]=B. Since B=CB⁒(P)⋉[B,P] by the hypothesis, we have 1=CB⁒(P)=NB⁒(P)=NH⁒(P)∩B. By the Frattini argument, this implies H=NH⁒(P)⁒Op′⁒(H)=NH⁒(P)⁒B=NH⁒(P)⋉B, and we are done.

(3) β‡’ (1). Let D be a nontrivial p-subgroup of G, and let H=SG⁒(D). By the hypothesis, there exist a Sylow p-subgroup P and a normal pβ€²-subgroup B of H such that H=NH⁒(P)⋉B. We assume that D≀P. Since B and PB are normal in H, DB is necessarily subnormal in H. Observe that H=SG⁒(D)=SH⁒(D), and therefore D=P=NH⁒(P). Consequently, D is self-normalizing in its subnormal closure H, as required. ∎

Proof of Theorem 1.4.

By Corollary 2.4, it suffices to prove that the class of π’žp-groups is closed under taking direct products. Let A and B be π’žp-groups, and let G=AΓ—B. Then A and B are p-solvable, and each of them has p-length at most 1. Let P be a p-subgroup of G, and write W=SG⁒(P). We work by induction on |A|+|B| to show that P is self-normalizing in W. Since Op′⁒(A) and Op′⁒(B) are π’žp-groups and P≀W≀Op′⁒(A)Γ—Op′⁒(B), by induction, we may assume that A=Op′⁒(A) and B=Op′⁒(B). In particular, A, B and G are p-nilpotent. Since W=SW⁒(P) is p-nilpotent, we have P∈Sylp⁒(W). Let

V∈Hallp′⁒(NW⁒(P)).

To see the self-normalization of P in W, it suffices to show that V=1.

Suppose that P⁒A=P⁒B=G. Since G=AΓ—B, we conclude that A, B and G are p-groups, and we are done. Consequently, we may assume that P⁒B<G. By the definition of the subnormal closure, we easily get

SG/B⁒(P⁒B/B)=SG⁒(P)⁒B/B.

Observe that

V⁒B/B≀W⁒B/B∩NG/B⁒(P⁒B/B)=NW⁒B/B⁒(P⁒B/B)=NSG/B⁒(P⁒B/B)⁒(P⁒B/B).

Since G/Bβ‰…A is a π’žp-group, we have V⁒B/B≀P⁒B/B, that is, V≀P⁒B. Now the p-nilpotency of PB implies V≀Op′⁒(B). Assume that P⁒A<G. The same arguments as above yield V≀Op′⁒(A), and hence V≀Op′⁒(A)∩Op′⁒(B)=1. Assume that P⁒A=G. Then B is a p-group, and thus V=1, as required. ∎

Proof of Theorem 1.5.

By Corollary 2.4, all subgroups and all quotient groups of G are π’žp-groups; also, G is p-solvable with lp⁒(G)≀1.

(1) Op′⁒(G) must be p-nilpotent since lp⁒(G)≀1. Then G=P⋉V, where P∈Sylp⁒(G) and V=Op′⁒(G). To see the solvability of Op′⁒(G), we may assume that P>1 and that G=Op′⁒(G) by induction. Let 1≀P1 be a maximal subgroup of P.

Assume that P1>1. Observe that

P1⁒V⊴G,Op′⁒(P1⁒V)⊴G,Op′⁒(G/Op′⁒(P1⁒V))=G/Op′⁒(P1⁒V).

Applying the inductive hypothesis to P1⁒V and G/Op′⁒(P1⁒V), we get that

Op′⁒(P1⁒V) and G/Op′⁒(P1⁒V)

are solvable, and so is G.

Assume that P1=1. Then P has order p. Since G=Op′⁒(G), we have

SG⁒(P)=G.

This implies NG⁒(P)=P and CV⁒(P)=1. Now G is a Frobenius group with V as its kernel. As is well known, V is nilpotent in this case [3, Theorem 9.5.1], and the result follows.

(2) Let x be an element of order p, and let H=SG⁒(γ€ˆx〉). Since H is p-solvable with lp⁒(H)=1 and SH⁒(γ€ˆx〉)=H, we see that H=γ€ˆx〉⋉V for a normal pβ€²-subgroup V of H. We claim that H has Fitting height at most 2. To prove the claim, we may assume that V>1. Since NH⁒(γ€ˆx〉)=γ€ˆx〉, H must be a Frobenius group with kernel V. This implies that V is nilpotent, and the claim holds. Now the subnormality of H in G yields H≀𝐅2⁒(G). Consequently, Ξ©p⁒(G)≀𝐅2⁒(G). ∎

3 π’ž~-groups and π’ž-groups

We denote by G𝒩 the nilpotent residual of G, that is, the smallest normal subgroup of G such that G/G𝒩 is nilpotent.

Lemma 3.1.

Let H be a subgroup of a finite solvable group G. Then SG⁒(H)=G if and only if G=H⁒GN.

Proof.

Assume that SG⁒(H)=G. Since H⁒G𝒩/G𝒩 is a subgroup of a nilpotent group G/G𝒩, H⁒G𝒩 is subnormal in G. This implies SG⁒(H)≀H⁒G𝒩, and hence G=H⁒G𝒩. Assume conversely that G=H⁒G𝒩. If SG⁒(H)<G, then there exists a maximal and normal subgroup M of G such that SG⁒(H)≀M. By the solvability of G, we have G𝒩≀M. This implies G=H⁒G𝒩≀M<G, a contradiction. Hence SG⁒(H)=G. ∎

Note that, for a subgroup H of an arbitrary finite group G, β€œSG⁒(H)=G” does imply β€œG=H⁒G𝒩”, but the converse is not true. For example, if G is a direct product of a nonabelian simple group and an abelian group H, then G=H⁒G𝒩, but SG⁒(H)=H<G.

Recall that a Carter subgroup of a finite group is a nilpotent and self-normalizing subgroup of the group. The following results about Carter subgroups are well known [2, Chapter VI, § 12].

Lemma 3.2.

For a finite solvable group G, the following hold:

  1. There exists a Carter subgroup of G, and all Carter subgroups of G are conjugate.

  2. If T is a Carter subgroup of G, then G=T⁒G𝒩.

Lemma 3.3.

Assume that G=H⋉V is a C~-group, where a nilpotent group H acts coprimely on a group V. Then C[V,H]⁒(H)=1.

Proof.

Note that, by Corollary 2.4, all subgroups and all quotient groups of G are also π’ž~-groups. We may assume that 1<H=AΓ—P, where 1<P=Op⁒(H) and A=Op′⁒(H) for some prime p. Since H⋉[V,H] is also a π’ž~-group, we may assume by induction that V=[V,H]. Clearly, [V,P] is normal in G. Since [V,H]=V and P centralizes V/[V,P], we conclude easily that

[V/[V,P],A⁒[V,P]/[V,P]]=V/[V,P].

Observe that (A⁒[V,P]/[V,P])⋉V/[V,P] also satisfies the hypothesis. Thus it follows by induction that

CV/[V,P]⁒(A⁒[V,P]/[V,P])=1.

Let x∈C[V,H]⁒(H). Since x centralizes A, we have

x⁒[V,P]∈CV/[V,P]⁒(A⁒[V,P]/[V,P]).

Then x∈A⁒[V,P], and

x∈C[V,H]⁒(H)∩A⁒[V,P]=C[V,H]⁒(H)∩[V,P]=C[V,P]⁒(H)≀C[V,P]⁒(P).

Note that G is a π’žp-group by Lemma 2.3. Considering the action of P on V and applying Theorem 1.3 (2), we get C[V,P]⁒(P)=1. Hence x=1, and we are done. ∎

Proof of Theorem 1.2.

By Lemma 2.3, we only need to show that a π’ž~-group must be a π’ž-group. Assume that G is a π’ž~-group. By Corollary 2.4, all subgroups and all quotient groups of G are π’ž~-groups; also, G is solvable with lp⁒(G)=1 for all primes p. Assume that G is not a π’ž-group, and let H<G be such that H is not self-normalizing in SG⁒(H). Let Ξ” be the set of elements of prime power order, in SG⁒(H), but not in H. Clearly, Ξ”β‰ βˆ…. In order to see a contradiction, we may assume that |H|+|G| is as small as possible.

By Corollary 2.4, H is not of prime power order, and hence |π⁒(H)|β‰₯2. Let U=G𝒩. Since all nilpotent groups are π’ž-groups, we have U>1. Let U/E be a chief factor of G, and assume that U/E is an elementary abelian q-group for some prime q. Now we work for a contradiction via several steps.

(1) SG⁒(H)=G, G=H⁒U, and H is nilpotent. By the minimality of |G|+|H|, we may assume that SG⁒(H)=G. Then G=H⁒U by Lemma 3.1. Suppose that H is not nilpotent, and let W be a Carter subgroup of H. Then W<H, and also H=W⁒H𝒩 by Lemma 3.2. Note that H/(H∩U)β‰…H⁒U/U=G/U is nilpotent; hence it follows that H𝒩≀H∩U. Now

G=H⁒U=W⁒H𝒩⁒U=W⁒U,

and thus G=SG⁒(W) by Lemma 3.1. The minimality of |G|+|H| yields that NG⁒(W)=W. Let xβˆˆΞ”, and take M=γ€ˆx〉⁒H. By the definition of Carter subgroup, Wm is also a Carter subgroup of H for every m∈M. By Lemma 3.2, it follows that Wm=Wh for some h∈H. Now

m∈NM⁒(W)⁒H and M=NM⁒(W)⁒H.

However, NM⁒(W)≀NG⁒(W)=W; this leads to the contradiction

M=NM⁒(W)⁒H=W⁒H=H.

Hence H is nilpotent.

(2) H is not maximal in G, and G/E=H⁒E/E⋉U/E, E>1. Assume that H is maximal in G. Since SG⁒(H)=G by (1), H is not normal in G. This implies NG⁒(H)=H, a contradiction. Since G=H⁒U, we have G/E=(H⁒E/E)⁒(U/E). If H⁒E=G, then G/Eβ‰…H/(H∩E) is nilpotent, against the fact that

U=G𝒩>E.

Hence H⁒E<G. As U/E is a chief factor of G, we have G/E=H⁒E/E⋉U/E. Moreover, since H is not maximal in G, we also get E>1.

(3) For every nontrivial normal subgroup N of G, H⁒N/N is a Carter subgroup of G/N, and Ξ”βŠ‚H⁒N. Since G/N is also a π’ž~-group and SG/N⁒(H⁒N/N)=G/N, we get by the minimality of |G|+|H| that H⁒N/N is self-normalizing in G/N. This implies that H⁒N/N is a Carter subgroup of G/N. Note that x∈NG⁒(H) implies x⁒N∈NG/N⁒(H⁒N/N), and it follows that Ξ”βŠ‚H⁒N.

(4) For some prime p, Ξ” consists only of p-elements, and

𝐅(G):=P∈Sylp(G).

Let N be a minimal normal subgroup of G, and assume that N is a p-group for some prime p. Then Ξ”βŠ‚H⁒N by (3). Let xβˆˆΞ”, and let D be a Hall pβ€²-subgroup of H. Obviously, D is a normal Hall pβ€²-subgroup of NH⁒N⁒(D), and this means that all pβ€²-elements of NH⁒N⁒(D) are contained in D. Observe that

x∈(H⁒N∩NG⁒(H))-HβŠ†(H⁒N∩NG⁒(D))-D=NH⁒N⁒(D)-D.

It follows that x is a p-element. Hence Ξ” consists only of p-elements. The arbitrariness of N also implies that all minimal normal subgroups of G are p-groups. In particular, 𝐅⁒(G) is a p-group. Since lp⁒(G)=1, we see that 𝐅⁒(G) is a Sylow p-subgroup of G.

(5) E is a p-group. Assume this is not true. Write N=Op⁒(E) and M=H⁒N. We have Ξ”βŠ‚M<G by (3) and (2). Clearly, M𝒩≀N because H is nilpotent, and N/M𝒩 is a pβ€²-group because N=Op⁒(N). Observe that, since 𝐅⁒(G) is a p-group and E is not a p-group, we get

1<Op⁒(N)≀M𝒩≀N◁M.

Write D=Op⁒(N). Since, by (3), H⁒D/D is a Carter subgroup of G/D, H⁒D/D is also a Carter subgroup of M/D. It follows by Lemma 3.2 that

M/D=(H⁒D/D)⁒(M/D)𝒩=(H⁒D/D)⁒(M𝒩/D)=H⁒M𝒩/D.

Thus M=H⁒M𝒩. Now Lemma 3.1 implies SM⁒(H)=M, and the minimality of |G|+|H| leads to the contradiction Ξ”βˆ©M=βˆ….

(6) Final contradiction. Recall that G=H⁒U, U/E is a q-chief factor of G and E is a p-group. Let A be a Hall {p,q}β€²-subgroup of H and let Q be a Sylow q-subgroup of G. Since G has a normal Sylow p-subgroup P by (4), we conclude easily that Q⁒P/E is a normal nilpotent subgroup of G/E. Observe that A=1 would imply that G/E=Q⁒P/E is nilpotent. Hence A>1. Write X=Q⁒P. Clearly, G=A⋉X and [X,A]≀U.

We claim that [X,A]=U. To prove this claim, we need only to show that U≀[X,A].

Assume that q=p. Then G=A⋉P and X=P. Since G/[X,A] is clearly nilpotent, we have [X,A]β‰₯U, and the claim follows.

Assume that qβ‰ p. Since E is a p-group, we get by the randomness of U/E that E is the unique maximal G-invariant subgroup of U. Consequently, either [X,A]=U or [X,A]≀E. Observe that if [X,A]≀E, then G/E is a direct product of a nilpotent {p,q}-subgroup X/E and a nilpotent {p,q}β€²-subgroup, against the non-nilpotency of G/E. Hence [X,A]=U, as claimed.

Now, considering the action of A on X, we get CU⁒(A)=1 by Lemma 3.3. This implies

NU⁒(H)≀NU⁒(A)=CU⁒(A)=1.

Since G=H⁒U, we get NG⁒(H)=H⁒(U∩NG⁒(H))=H⁒NU⁒(H)=H. This is the final contradiction, and the proof is complete. ∎


Communicated by Christopher W. Parker


Award Identifier / Grant number: BK20161265

Award Identifier / Grant number: 11871011

Award Identifier / Grant number: 11671063

Funding statement: Project supported by the NSF of Jiangsu Province (No. BK20161265) and the NSF of China (Nos. 11871011, 11671063).

Acknowledgements

The author is grateful to the referee for his/her valuable comments.

References

[1] A. Ballester-Bolinches, J. Cossey and Y. Li, On a class of finite soluble groups, J. Group Theory 21 (2018), no. 5, 839–846. 10.1515/jgth-2018-0015Search in Google Scholar

[2] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar

[3] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups. An Introduction, Springer, Berlin, 2004. 10.1007/b97433Search in Google Scholar

Received: 2019-01-13
Revised: 2019-03-25
Published Online: 2019-05-07
Published in Print: 2019-09-01

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