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
From the journal Journal of Group Theory

## Abstract

For a given prime p, a finite group G is said to be a π~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~p-groups, show that the class of π~p-groups is a subgroup-closed formation and that Opβ²β’(G) is a solvable p-nilpotent group for every π~p-group G. We also prove that if a finite group G is a π~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 SGβ’(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-group (respectively, a πp-group) if every subgroup (respectively, every p-subgroup) H of G is self-normalizing in SGβ’(H).

2. G is said to be a π~-group (respectively, a π~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p-group G is p-solvable with lpβ’(G), the p-length of G, at most 1. In particular, all π-groups are solvable. Clearly, a π-group is necessarily a πp 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p-group if and only if G is a π~p-group.

2. G is a π-group if and only if G is a π~-group.

For a given prime p, it is shown in [1, Theorem 8] that G is a πp-group if and only if G is p-solvable with lpβ’(G)β€1, and COpβ’(SGβ’(H))β’(H)=1 for all subgroups H of G. The following descriptions of πp-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p-group.

2. G is p-solvable with lpβ’(G)β€1, and if a p-subgroup P of G acts on a pβ²-subgroup V of G, then V=CVβ’(P)β[V,P], that is, C[V,P]β’(P)=1.

3. Every subgroup H of G can be expressed as H=NHβ’(P)βB, where P is a Sylow p-subgroup of H and B is a normal pβ²-subgroup of H.

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

This implies CBβ’(P)=1 and B=[B,P].

Combining Theorems 1.2 and 1.3, we get some descriptions of π-groups. As shown in [1, Theorem A], the class of π-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 πiβ’(G) be the ith ascending Fitting subgroup of G, that is, π0β’(G)=1, π1β’(G)=πβ’(G) and πi+1β’(G)/πiβ’(G)=πβ’(G/πiβ’(G)). The Fitting height of a solvable group G is the smallest integer k such that πkβ’(G)=G. For a given prime p, we denote by Ξ©pβ’(G) the characteristic subgroup of G generated by all elements of order p. Clearly, Ξ©pβ’(G)β€Opβ²β’(G).

## Theorem 1.5.

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

1. Opβ²β’(G) is solvable and p-nilpotent.

2. Ξ©pβ’(G) has Fitting height at most 2.

Let G be a π-group. By Theorem 1.5β(2), we get that Ξ©β’(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~p-groups and πp-groups

In this section, we will investigate π~p-groups and πp-groups. For a subgroup D of G, it is easy to see that SGβ’(D)=SSGβ’(D)β’(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~p-group,

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

3. G is p-solvable with lpβ’(G)β€1.

## Proof.

(1) Let H be a subgroup of G, and let P be a cyclic p-subgroup of H. Since SGβ’(P) is subnormal in G, we have that SGβ’(P)β©H is subnormal in H. This implies SHβ’(P)β€SGβ’(P)β©Hβ€SGβ’(P). Now the self-normalization of P in SGβ’(P) yields NSHβ’(P)β’(P)=P. Hence H is a π~p-group.

(2) Let Nβ΄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/Nβ’(U/N). Assume that SG/N(U/N):=H/N<G/N. Since H is also a π~p-group by (1), we get by induction that H/N is again a π~p-group. Thus NH/Nβ’(U/N)=U/N, as required. Therefore, we may assume that SG/Nβ’(U/N)=G/N.

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

NSGβ’(P)β’(P)=P.

Suppose that SGβ’(P)=G. Then NGβ’(P)=P, so PβSylpβ’(G). Let us investigate NGβ’(Pβ’N). Clearly, NGβ’(P)β€NGβ’(Pβ’N), and thus NGβ’(P)=NNGβ’(Pβ’N)β’(P). By the Frattini argument and the normality of PN in NGβ’(Pβ’N), we get

NGβ’(Pβ’N)=Pβ’Nβ’NNGβ’(Pβ’N)β’(P)=Pβ’Nβ’NGβ’(P)=Pβ’N.

Clearly, NG/Nβ’(Pβ’N/N)=NGβ’(Pβ’N)/N. This yields the required result

NG/Nβ’(U/N)=NG/Nβ’(Pβ’N/N)=NGβ’(Pβ’N)/N=Pβ’N/N=U/N.

Suppose that SGβ’(P)<G. Let M be a maximal normal subgroup of G such that SGβ’(P)β€M. Since U/Nβ€Mβ’N/Nβ΄G/N and SG/Nβ’(U/N)=G/N, we have G/N=Mβ’N/N. Now the minimal normality of N implies G=MΓN. Note that M is a π~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~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 NGβ’(P)=P by the hypothesis, we have PβSylpβ’(G) and NGβ’(P)=CGβ’(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 lpβ’(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=Opβ’(G) and Ξ¦β’(G)=1;

3. G=MβE for a maximal subgroup M of G, and M=PβK, where M=Opβ²β’(M), K=Opβ²β’(K) and 1<PβSylpβ’(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 lpβ’(A1)β€1 and lpβ’(A2)β€1 by induction, we have

lpβ’(G)=lpβ’(A1β’A2)=maxβ‘{lpβ’(A1),lpβ’(A2)}β€1,

Now it is easy to see that G is the subnormal closure of P in G. Then we have NGβ’(P)=P by the hypothesis. However, P<PβEβSylpβ’(G), a contradiction. Thus lpβ’(G)β€1. β

## Lemma 2.2.

Let D and V be subgroups of G with (|D|,|V|)=1. Assume that G=DβV and that D is self-normalizing in SGβ’(D). Then

C[V,D]β’(D)=1,V=CVβ’(D)β[V,D],G=NGβ’(D)β[V,D]βπππβNGβ’(D)=DΓCVβ’(D).

## Proof.

Since D acts coprimely on V, we have V=[V,D]β’CVβ’(D) by [3, Theorem 8.2.7]. Clearly, [V,D] and Dβ[V,D] are normal in G. Note that (see [3, Theorem 8.2.7]) [V,D,D]=[V,D]. It follows that SGβ’(D)=Dβ[V,D].

Clearly, C[V,D]β’(D)=N[V,D]β’(D). Since NSGβ’(D)β’(D)=D by the hypothesis, we get

This implies V=[V,D]β’CVβ’(D)=CVβ’(D)β[V,D]. Also, we get

and G=NGβ’(D)β[V,D]. β

## 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ifβ part. Let G be a π~p-group, and let P be a p-subgroup of G. To see that P is self-normalizing in SGβ’(P), we may assume that P is not cyclic and that SGβ’(P)=G by induction. Notice that G is p-solvable with lpβ’(G)=1. Now SGβ’(P)=G implies

G=Opβ²β’(G)βandβG=PβV

for a normal pβ²-subgroup V of G. We only need to show that CVβ’(P)=1 since NVβ’(P)=CVβ’(P) and NGβ’(P)=Pβ’NVβ’(P).

Let xβCVβ’(P), and let D be a maximal subgroup of P. Clearly, both [V,D] and Dβ’[V,D] are normal in G. Note that G/Dβ’[V,D]=Opβ²β’(G/Dβ’[V,D]). Hence

G/Dβ’[V,D]=SG/Dβ’[V,D]β’(Pβ’[V,D]/Dβ’[V,D]).

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

xβPβ’[V,D],

Observe that DβV is also a πp-group by induction; therefore, by Lemma 2.2, we get C[V,D]β’(D)=1. Hence x=1, CVβ’(P)=1, and the proof is complete. β

## Corollary 2.4.

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

1. Assume that G is a πp-group. Then all subgroups and all quotient groups of G are πp-groups; also, G is p-solvable with lpβ’(G)β€1.

2. Assume that G is a π~-group. Then all subgroups and all quotient groups of G are π~-groups, G is solvable with lrβ’(G)β€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. β

## Proof of Theorem 1.3.

(1) β (2). Assume that G is a πp-group. By Corollary 2.4, G is p-solvable with lpβ’(G)β€1. Let P and V be a p-subgroup and a pβ²-subgroup, respectively, of G such that P acts on V. Since PβV is again a πp-group by Corollary 2.4, the required result follows by Lemma 2.2.

(2) β (3). Let Hβ€G, and let PβSylpβ’(H). We may assume that P>1. Clearly, H is p-solvable with lpβ’(H)=1. Then Opβ²β’(H)=PβB for a normal pβ²-subgroup B of H. Clearly, [B,P]=B. Since B=CBβ’(P)β[B,P] by the hypothesis, we have 1=CBβ’(P)=NBβ’(P)=NHβ’(P)β©B. By the Frattini argument, this implies H=NHβ’(P)β’Opβ²β’(H)=NHβ’(P)β’B=NHβ’(P)βB, and we are done.

(3) β (1). Let D be a nontrivial p-subgroup of G, and let H=SGβ’(D). By the hypothesis, there exist a Sylow p-subgroup P and a normal pβ²-subgroup B of H such that H=NHβ’(P)βB. We assume that Dβ€P. Since B and PB are normal in H, DB is necessarily subnormal in H. Observe that H=SGβ’(D)=SHβ’(D), and therefore D=P=NHβ’(P). Consequently, D is self-normalizing in its subnormal closure H, as required. β

## Proof of Theorem 1.4.

By Corollary 2.4, it suffices to prove that the class of πp-groups is closed under taking direct products. Let A and B be πp-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=SGβ’(P). We work by induction on |A|+|B| to show that P is self-normalizing in W. Since Opβ²β’(A) and Opβ²β’(B) are πp-groups and Pβ€Wβ€Opβ²β’(A)ΓOpβ²β’(B), by induction, we may assume that A=Opβ²β’(A) and B=Opβ²β’(B). In particular, A, B and G are p-nilpotent. Since W=SWβ’(P) is p-nilpotent, we have PβSylpβ’(W). Let

VβHallpβ²β’(NWβ’(P)).

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

Suppose that Pβ’A=Pβ’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 Pβ’B<G. By the definition of the subnormal closure, we easily get

SG/Bβ’(Pβ’B/B)=SGβ’(P)β’B/B.

Observe that

Since G/BβA is a πp-group, we have Vβ’B/Bβ€Pβ’B/B, that is, Vβ€Pβ’B. Now the p-nilpotency of PB implies Vβ€Opβ²β’(B). Assume that Pβ’A<G. The same arguments as above yield Vβ€Opβ²β’(A), and hence Vβ€Opβ²β’(A)β©Opβ²β’(B)=1. Assume that Pβ’A=G. Then B is a p-group, and thus V=1, as required. β

## Proof of Theorem 1.5.

By Corollary 2.4, all subgroups and all quotient groups of G are πp-groups; also, G is p-solvable with lpβ’(G)β€1.

(1) Opβ²β’(G) must be p-nilpotent since lpβ’(G)β€1. Then G=PβV, where PβSylpβ’(G) and V=Opβ²β’(G). To see the solvability of Opβ²β’(G), we may assume that P>1 and that G=Opβ²β’(G) by induction. Let 1β€P1 be a maximal subgroup of P.

Assume that P1>1. Observe that

P1β’Vβ΄G,Opβ²β’(P1β’V)β΄G,Opβ²β’(G/Opβ²β’(P1β’V))=G/Opβ²β’(P1β’V).

Applying the inductive hypothesis to P1β’V and G/Opβ²β’(P1β’V), we get that

Opβ²β’(P1β’V)βandβG/Opβ²β’(P1β’V)

are solvable, and so is G.

Assume that P1=1. Then P has order p. Since G=Opβ²β’(G), we have

SGβ’(P)=G.

This implies NGβ’(P)=P and CVβ’(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=SGβ’(γxγ). Since H is p-solvable with lpβ’(H)=1 and SHβ’(γxγ)=H, we see that H=γxγβ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 NHβ’(γxγ)=γxγ, 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 Hβ€π2β’(G). Consequently, Ξ©pβ’(G)β€π2β’(G). β

## 3 π~-groups and π-groups

We denote by Gπ© the nilpotent residual of G, that is, the smallest normal subgroup of G such that G/Gπ© is nilpotent.

## Lemma 3.1.

Let H be a subgroup of a finite solvable group G. Then SGβ’(H)=G if and only if G=Hβ’GN.

## Proof.

Assume that SGβ’(H)=G. Since Hβ’Gπ©/Gπ© is a subgroup of a nilpotent group G/Gπ©, Hβ’Gπ© is subnormal in G. This implies SGβ’(H)β€Hβ’Gπ©, and hence G=Hβ’Gπ©. Assume conversely that G=Hβ’Gπ©. If SGβ’(H)<G, then there exists a maximal and normal subgroup M of G such that SGβ’(H)β€M. By the solvability of G, we have Gπ©β€M. This implies G=Hβ’Gπ©β€M<G, a contradiction. Hence SGβ’(H)=G. β

Note that, for a subgroup H of an arbitrary finite group G, βSGβ’(H)=Gβ does imply βG=Hβ’Gπ©β, 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=Hβ’Gπ©, but SGβ’(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, Β§β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=Tβ’Gπ©.

## Lemma 3.3.

Assume that G=HβV is a C~-group, where a nilpotent group H acts coprimely on a group V. Then C[V,H]β’(H)=1.

## Proof.

Note that, by Corollary 2.4, all subgroups and all quotient groups of G are also π~-groups. We may assume that 1<H=AΓP, where 1<P=Opβ’(H) and A=Opβ²β’(H) for some prime p. Since Hβ[V,H] is also a π~-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],Aβ’[V,P]/[V,P]]=V/[V,P].

Observe that (Aβ’[V,P]/[V,P])βV/[V,P] also satisfies the hypothesis. Thus it follows by induction that

CV/[V,P]β’(Aβ’[V,P]/[V,P])=1.

Let xβC[V,H]β’(H). Since x centralizes A, we have

xβ’[V,P]βCV/[V,P]β’(Aβ’[V,P]/[V,P]).

Then xβAβ’[V,P], and

Note that G is a πp-group by Lemma 2.3. Considering the action of P on V and applying Theorem 1.3β(2), we get C[V,P]β’(P)=1. Hence x=1, and we are done. β

## Proof of Theorem 1.2.

By Lemma 2.3, we only need to show that a π~-group must be a π-group. Assume that G is a π~-group. By Corollary 2.4, all subgroups and all quotient groups of G are π~-groups; also, G is solvable with lpβ’(G)=1 for all primes p. Assume that G is not a π-group, and let H<G be such that H is not self-normalizing in SGβ’(H). Let Ξ be the set of elements of prime power order, in SGβ’(H), but not in H. Clearly, Ξβ β. 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 |Οβ’(H)|β₯2. Let U=Gπ©. Since all nilpotent groups are π-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) SGβ’(H)=G, G=Hβ’U, and H is nilpotent. By the minimality of |G|+|H|, we may assume that SGβ’(H)=G. Then G=Hβ’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=Wβ’Hπ© by Lemma 3.2. Note that H/(Hβ©U)βHβ’U/U=G/U is nilpotent; hence it follows that Hπ©β€Hβ©U. Now

and thus G=SGβ’(W) by Lemma 3.1. The minimality of |G|+|H| yields that NGβ’(W)=W. Let xβΞ, and take M=γxγβ’H. By the definition of Carter subgroup, Wm is also a Carter subgroup of H for every mβM. By Lemma 3.2, it follows that Wm=Wh for some hβH. Now

mβNMβ’(W)β’HβandβM=NMβ’(W)β’H.

However, NMβ’(W)β€NGβ’(W)=W; this leads to the contradiction

M=NMβ’(W)β’H=Wβ’H=H.

Hence H is nilpotent.

(2) H is not maximal in G, and G/E=Hβ’E/EβU/E, E>1. Assume that H is maximal in G. Since SGβ’(H)=G by (1), H is not normal in G. This implies NGβ’(H)=H, a contradiction. Since G=Hβ’U, we have G/E=(Hβ’E/E)β’(U/E). If Hβ’E=G, then G/EβH/(Hβ©E) is nilpotent, against the fact that

Hence Hβ’E<G. As U/E is a chief factor of G, we have G/E=Hβ’E/EβU/E. Moreover, since H is not maximal in G, we also get E>1.

(3) For every nontrivial normal subgroup N of G, Hβ’N/N is a Carter subgroup of G/N, and ΞβHβ’N. Since G/N is also a π~-group and SG/Nβ’(Hβ’N/N)=G/N, we get by the minimality of |G|+|H| that Hβ’N/N is self-normalizing in G/N. This implies that Hβ’N/N is a Carter subgroup of G/N. Note that xβNGβ’(H) implies xβ’NβNG/Nβ’(Hβ’N/N), and it follows that ΞβHβ’N.

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

π(G):=Pβ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βHβ’N by (3). Let xβΞ, and let D be a Hall pβ²-subgroup of H. Obviously, D is a normal Hall pβ²-subgroup of NHβ’Nβ’(D), and this means that all pβ²-elements of NHβ’Nβ’(D) are contained in D. Observe that

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, πβ’(G) is a p-group. Since lpβ’(G)=1, we see that πβ’(G) is a Sylow p-subgroup of G.

(5) E is a p-group. Assume this is not true. Write N=Opβ’(E) and M=Hβ’N. We have ΞβM<G by (3) and (2). Clearly, Mπ©β€N because H is nilpotent, and N/Mπ© is a pβ²-group because N=Opβ’(N). Observe that, since πβ’(G) is a p-group and E is not a p-group, we get

Write D=Opβ’(N). Since, by (3), Hβ’D/D is a Carter subgroup of G/D, Hβ’D/D is also a Carter subgroup of M/D. It follows by Lemma 3.2 that

Thus M=Hβ’Mπ©. Now Lemma 3.1 implies SMβ’(H)=M, and the minimality of |G|+|H| leads to the contradiction Ξβ©M=β.

(6) Final contradiction. Recall that G=Hβ’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 Qβ’P/E is a normal nilpotent subgroup of G/E. Observe that A=1 would imply that G/E=Qβ’P/E is nilpotent. Hence A>1. Write X=Qβ’P. Clearly, G=AβX and [X,A]β€U.

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

Assume that q=p. Then G=Aβ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]β€E. Observe that if [X,A]β€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 CUβ’(A)=1 by Lemma 3.3. This implies

NUβ’(H)β€NUβ’(A)=CUβ’(A)=1.

Since G=Hβ’U, we get NGβ’(H)=Hβ’(Uβ©NGβ’(H))=Hβ’NUβ’(H)=H. This is the final contradiction, and the proof is complete. β

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.

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