Boyan WEIHaipeng QUYanfeng LUO

1 Introduction

It is well-known that thefinite p-groups are fundamental tools in understanding the structure offinite groups and in the classification offinite simple groups.After the classification offinite simple groups was completed,the study of p-groups has attracted much attention and thefinite p-groups have been investigated intensively over the past decades.However,the complex behaviour offinite p-groups is well-known and it prevents any attempt at a general classification.In fact,only the p-groups of order at most p7have been completely classified for a general prime p(see[12]for detail).This can be extended(see[5])to the groups of order less than or equal to p10in the case p=2.This situation has naturally led to restricting the study of p-groups to special families.

The enumeration problem of p-groups is important in the study offinite p-groups,which includes two aspects:One is to study the number of subgroups,elements and subsets offinite p-groups.A classical result of Kulakoff[11]asserts:For a prime p＞2 and a group G of order pn,sk(G)≡1 or 1+p(mod p2),where sk(G)denotes the number of subgroups of order pkof G.Along Kulakoff’s idea,many scholars investigated the possible case of sk(G)(mod p3)(see[2,7,10,15,17–18]).The other is to study the structure or properties offinite p-groups by means of the number of subgroups.Fan[8]characterizedfinite elementary abelian p-groups in term of sk(G).Moreover,Qu[13]characterizedfinite non-elementary abelian p-groups whose number of subgroups of possible order is maximal.More results about the enumeration of subgroups can be referenced to[1,3,14,16,19].In this paper,we continue to characterize the structure of p-groups by the enumeration of subgroups.

In the enumeration of subgroups,a fundamental theorem is the enumeration principle given by P.Hall in[9].The theorem is stated as follows.

Theorem 1.1(P.Hall Enumeration Principle)

whereGis afinitep-group,d=d(G),s(M)denotes the number of subgroups contained in a subgroupMofG,Sidenotes the set consisting of all major subgroups(in the sense that subgroups containingΦ(G))of indexpiofG.

We observe that in the enumeration principle,P.Hall introduced a kind of important subgroups,major subgroups,which play important roles in the study offinite p-groups.Moreover,for a group G of order pn,P.Hall Enumeration Principle gives the relationship between the number of major subgroups of index pkand the number of subgroups of index pk.For convenience,a subgroup H of index pkis called a k-maximal subgroup of G.A k-maximal subgroup H is called a k-major subgroup of G if H is major;otherwise,H is called a non-major k-maximal subgroup of G.Clearly,if H is a k-major subgroup of G,then 1≤k≤d=d(G).

Let γk(G)and δk(G)denote the number of all k-major subgroups of G and the number of non-major k-maximal subgroups of G,respectively.Then δk(G)=sn−k(G) − γk(G)for k=1,2,···,d.Moreover,by the corresponding theorem,we have that γk(G)is equal to the number of k-maximal subgroups of an elementary abelian group of order pd.P.Hall[9]proved that

It means that δk(G)≡ 0(mod p)for 1 ≤ k ≤ d.

A natural question is:How does δk(G)influence the structure of a p-group G?

Obviously,δ1(G)=0 is true for any finite p-group G.So we get nothing in this case.It is easy to see that for a finite p-group G,if δk(G)=0 for some k ∈ {1,···,d},then δt(G)=0 for all t∈ {1,···,k}.This implies that if k is a smaller positive integer,then there are many p-groups with δk(G)=0.So we pay our attention to the cases of k=d and k=d − 1.In this paper,we study thefinite p-groups with few non-major k-maximal subgroups.We prove that if 0 ＜ δk(G)＜ p2for some k ∈ {1,2,···,d},then k=d and δd(G)=p.Moreover,we classify allfinite p-groups with δd(G)=0,δd−1(G)=0,δd(G)=p and δd(G)=p2,respectively.

2 Preliminaries

In this section,we give some notation and background material of this article.The reader is referred to[3]for any undefined notation and terminology in this article.

Let G be afinite p-group.We use exp(G),c(G)and d(G)to denote the exponent,the nilpotency class of G and the number of elements in a minimal generator-system of G,respectively.G is said to be of maximal class if|G|=pn(n≥3)and c(G)=n−1.Denote by Cpnand Cmpnthe cyclic group of order pnand the direct product of m cyclic groups of order pn,respectively.M⋖G means that M is a maximal subgroup of G and H G means that H is isomorphic to a subgroup of G.Let H and K be two subgroups of G.We use H∗K to denote the central product of H and K.

For each positive integer i,we define

Let

To end this section,we list some known results which will be used in the sequel.

Lemma 2.1(see[9])LetGbe afinitep-group andd=d(G).Thenδk(G) ≡ 0(mod pd−k+1)for1≤ k≤ d.

Lemma 2.2(see[3])Assume thatGis a group of orderpn,0≤k≤n.Thensk(G)≡1(mod p).

Lemma 2.3(see[3])LetGbe a group of orderpn.IfGis neither cyclic nor a2-group of maximal class,thensk(G)≡1+p(mod p2)for1≤k＜n.

Lemma 2.4(see[3])LetGbe a group of orderpn.

(1)If1＜m＜nandsm(G)=1,thenGis cyclic;

(2)Ifs1(G)=1,thenGis cyclic or generalized quaternion.

Lemma 2.5(see[3])LetGbe ap-group with|G′|=p.Then

whereA1,A2,···,Asare minimal nonabelian,soG/Z(G)is elementary abelian of even rank.In particular,ifG/G′is elementary abelian,then

LetE=A1∗A2∗···∗As.ThenEis extraspecial andG=EZ(G).

Lemma 2.6(see[3])LetGbe afinitep-group.Ifc(G)＜ p,thenGis regular.

The following lemmas are the classifications offinite p-groups with some properties,which will be used in the sequel.

Lemma 2.7(see[4])LetGbe a nonabelianp-group withd(G)=3.Suppose that all maximal subgroups ofGare generated by two elements.Ifc(G)=2,then|G|≤p6and one of the following holds:

(1)G=Mp(2,1)∗ Cp2Mp(1,1,1)∗ Cp2;

(2)G=Q8×C2;

(3)G= 〈a,b,c|a4=b4=1,c2=a2b2,[a,b]=1,[a,c]=a2,[b,c]=a2b2〉;

(4)G= 〈a,b,c|a4=b4=c2=1,[a,b]=1,[a,c]=a2b2,[b,c]=a2〉;

(5)Ifp ＞ 2,thenG= 〈a,b,c,x,y|ap=bp2=cp2=1,[b,c]=1,[a,b]=x,[a,c]=y,bp=xαyβ,cp=xγyδ,xp=yp=[b,x]=[b,y]=[c,x]=[c,y]=1〉,where4βγ +(δ− α)2is a quadratic non-residue modp;

(6)|G|=26,G=〈a,b,c|a4=b4=c4=1,[a,b]=c2,[a,c]=b2c2,[b,c]=a2b2,[a2,b]=[a2,c]=[b2,a]=[b2,c]=[c2,a]=[c2,b]=1〉.

As a direct consequence of[4,Theorems 70.2,70.4–70.5],we have the following lemma.

Lemma 2.8Assume thatGis a nonabelianp-group withd(G)=3and all maximal subgroups ofGare generated by two elements.Ifc(G)＞2,thenGis one of the following nonisomorphic groups:

(1)〈a,b,c,d|a4=b4=c4=d2=[a,d]=[b,d]=[c,d]=1,[a,b]=c2,[a,c]=b2c2,[b,c]=a2b2,[a2,b]=[a2,c]=[b2,c]=[c2,a]=d,[b2,a]=1,[c2,b]=1〉;

(2)H=G/G4=〈a,b,c,d,e,f|a4=b4=c4=d2=e2=f2=def=[a,d]=[b,d]=[c,d]=[a,e]=[b,e]=[c,e]=1,[a,b]=c2d∈,[a,c]=b2c2d∈,[b,c]=a2b2,[a2,b]=[b2,c]=[c2,a]=f,[a2,c]=d,[b2,a]=[c2,b]=e〉,where∈=0,1.

Lemma 2.9(see[6])Suppose thatGis a group of order26and all2-maximal subgroups ofGare metacyclic.IfGhas a maximal subgroup which is not metacyclic,thenG＞H=〈a,b,c|a4=b4=1,c2=a2b2,[a,b]=1,[a,c]=a2,[b,c]=a2b2〉andGis one of the following groups:

(1) 〈H,d|d2=a2,[a,d]=a2b2,[b,d]=b2,[c,d]=1〉;

(2) 〈H,d|d2=b2,[a,d]=a2,[b,d]=a2b2,[c,d]=a−1〉;

(3) 〈H,d|d2=b,[a,d]=b2,[b,d]=1,[c,d]=a−1〉.

Lemma 2.10(see[3])LetGbe a nonabelian group of orderpnwith cyclic maximal subgroup.ThenGis isomorphic to one of the following groups:

(1)Mp(n−1,1),wheren≥4ifp=2;

(2)D2n= 〈a,b|a2n−1=1,b2=1,[a,b]=a−2〉(n ≥ 3),the dihedral group;

(3)Q2n= 〈a,b|a2n−1=1,b2=a2n−2,[a,b]=a−2〉(n ≥ 3),the generalized quaternion group;

(4)SD2n= 〈a,b|a2n−1=1,b2=1,[a,b]=a−2+2n−2〉(n ≥ 4),the semidihedral group.

3 The Classification for the Cases δd(G)=0 and δd−1(G)=0

In this section,we classify allfinite p-groups G with δd(G)=0 and δd−1(G)=0,respectively.Throughout this section,we always assume that d=d(G)＞1.

We first consider the case δd(G)=0.The next theorem gives the classification of allfinite p-groups G with δd(G)=0.

Theorem 3.1LetGbe a group of orderpnandd=d(G)＞ 1.Thenδd(G)=0if and only if eitherGQ8orG.

Proof IfGQ8or G,then it is easy to verify that δd(G)=0.

Conversely,suppose that δd(G)=0.Then G contains only one d-maximal subgroup Φ(G)and sn−d(G)=1+δd(G)=1.If n−d ＞ 1,then G is cyclic by Lemma 2.4.This contradicts the assumption d ＞1.Thus n−d≤ 1.If n−d=0,then G.If n−d=1,then Gby Lemma 2.4.Thus d=2 and n=3.That is,GQ8.

To classify allfinite p-groups with δd−1(G)=0,we need the following lemmas which give some properties of a finite p-group G with δk(G)=0 for some k ∈ {2,···,d}.

Lemma 3.1LetGbe afinitep-group andka positive integer such that2≤k≤d.Then the following conditions are equivalent:

(1)δk(G)=0.

(2)Φ(H)=Φ(G)for every(k−1)-maximal subgroupHofG.

(3)d(H)=d−(k−1)for every(k−1)-maximal subgroupHofG.

Proof(1)(2)Suppose that(1)holds.Let H be any(k − 1)-maximal subgroup of G.Then every maximal subgroupof H is a k-maximal subgroup of G.Since δk(G)=0,we have Φ(G)≤for all maximal subgroupsof H.It follows that

Since every subgroup of a finite p-group is subnormal,we have Φ(H) ≤ Φ(G).Therefore Φ(H)= Φ(G)and(2)holds.

(2)⇒(3)It is obvious.

Lemma 3.3LetGbe afinitep-group.Ifδk(G)=0for some positive integerk ∈{2,3,···,d},thenδk−1(M)=0for every maximal subgroupMofG.

ProofObserve that δ2(G)=0.Then,by Lemma 3.1(2),Φ(M)= Φ(G)for every maximal subgroup M of G.Notice that every(k−1)-maximal subgroup of M is a k-maximal subgroup of G.It follows from the fact that δk(G)=0 that all(k − 1)-maximal subgroups of M contain Φ(G)= Φ(M).Therefore δk−1(M)=0.

Now we are ready for the classification of allfinite p-groups G with δd−1(G)=0.

Theorem 3.2LetGbe afinitep-group withd ＞ 2.Thenδd−1(G)=0if and only if one of the following holds:

ProofIt is routine matter to verify that each group listed in the theorem satisfies δd−1(G)=0.

Conversely,suppose that G is a finite p-group with d ＞ 2 such that δd−1(G)=0.Then G′= Φ(G)by Lemma 3.2.Since d ≥ 3 and δd−1(G)=0,we have δ2(G)=0.If G is abelian,then G is elementary abelian by Lemma 3.2 and(1)holds.Suppose that G is nonabelian.There are three cases to be considered.

Case 1d=3.Then δ2(G)=0.It follows from Lemma 3.1(3)that all maximal subgroups of G are generated by two elements.It follows from Lemmas 2.7–2.8 that G is one of the groups listed in Lemmas 2.7–2.8.Hence(2)holds.

Case 2d=4.Then δ3(G)=0 and δ2(G)=0.Let M be any maximal subgroup of G.Then d(M)=3 by Lemma 3.1(3).In addition,δ2(M)=0 by Lemma 3.3.Hence M is one of the groups listed in Lemmas 2.7–2.8 by Case 1.

We claim that p=2.

Otherwise,let G be a counterexample of minimal order.Then G′= Φ(G)by Lemma 3.2.Let N be a normal subgroup of G such that|G′:N|=p.Then G/N is also a counterexample.By the minimality of G,we must have N=1.Hence|G|=p5,|G′|=|Φ(G)|=p and

Suppose that|G|=2n.Then n≥5.

(i)Assume|G|=25.Then|G′|=|Φ(G)|=2 since G′= Φ(G).It follows from Lemma 2.5 that G is isomorphic to one of the following groups:

If G≇D8∗Q8,then G has an elementary abelian subgroup H of order 8.By Lemma 3.1(2),Φ(G)= Φ(H)=1,a contradiction.Therefore GD8∗Q8.

(ii)Assume|G|=26.Then|G′|=|Φ(G)|=4.It follows that M is the group listed in Lemma 2.7(3)or(4).

We claim that every maximal subgroup of G is isomorphic to the group listed in Lemma 2.7(3).

Suppose to contrary that there exists a maximal subgroup M of G which is isomorphic to the group listed in Lemma 2.7(4).Let K be a subgroup of M such that KC32and M′≤ K.Notice that|M′|=4 and M′=G′.Then K G.If K ＜ CG(K),then there exists L≤CG(K)such that K⋖L.So L is abelian,|L|=24and d(L)≥d(K)=3.Since δ3(G)=0,we have d(L)=2 by Lemma 3.1(3),a contradiction.So CG(K)=K and hence G/K Aut(K)GL(3,2).Notice that the Sylow 2-subgroup of GL(3,2)is isomorphic to D8.It follows that G/KD8.However,since Φ(G)=G′≤ K,we have G/K is elementary abelian,a contradiction.Therefore every maximal subgroup of G is isomorphic to the group listed in Lemma 2.7(3).

By a direct calculation,we may show that every maximal subgroup of the group listed in Lemma 2.7(3)is isomorphic toor M2(2,2).So all 2-maximal subgroups of G are metacyclic.Hence G is isomorphic to one of the groups listed in Lemma 2.9.Notice that d=4.Hence GP.

(iii)We claim that|G|=25or 26.

Otherwise,suppose n≥7 and let G be a counterexample of minimal order.Then G′=Φ(G)by Lemma 3.2.Let N be a normal subgroup of G such that|G:N|=27and N≤Φ(G).Then G/N is also a counterexample.By the minimality of G,we must have N=1.Hence|G|=27,|G′|=|Φ(G)|=8 and M is isomorphic to the group listed in Lemma 2.7(6).

Now assume

Let G=G/〈a2〉.Then|G|=26and δ3(G)=0.So every maximal subgroup H of G should be isomorphic to the group listed in Lemma 2.7(3).In particular,each element in HΦ(H)is of order 4.But∈MΦ(M)is an involution,a contradiction.

Case 3d ≥ 5.In this case,we show that there is no finite p-group G such that δd−1(G)=0.

Otherwise,let G be a counterexample of minimal order.Let H be a(d−5)-maximal subgroup of G.Then d(H)=5.Take N ⊳ H such that|Φ(H):N|=p.Then H/N is also a counterexample.By the minimality of G,we have N=1 and H=G.So|G|=p6and|G′|=|Φ(G)|=p.Assume that M is an arbitrary maximal subgroup of G.Then d(M)=4 and δ3(M)=0.Hence p=2 and MD8∗ Q8by Case 2.It follows from Lemma 2.5 that

Thus G has an elementary abelian subgroup H of order 8.By Lemma 3.1(2),Φ(G)= Φ(H)=1.This is a contradiction.

4 The Classification for the Cases δd(G)=p and δd(G)=p2

In this section,we classify all finite p-groups G with δd(G)=p and δd(G)=p2,where d=d(G)＞1.

We first consider the case δd(G)=p2.The next theorem gives the classification of allfinite p-groups G with δd(G)=p2.

Theorem 4.1LetGbe a noncyclic group of orderpn.Thenδd(G)=p2if and only ifGis a2-group of maximal class andGQ8.

ProofIf G is a 2-group of maximal class and GQ8,then d=2,and δ2(G)=22by counting the 2-maximal subgroups of G.Conversely,assume that δd(G)=p2.Then G is not elementary abelian and so 1≤n−d＜n.If G is not a 2-group of maximal class,then sn−d(G)=1+δd(G) ≡ 1+p(mod p2)by Lemma 2.3.This contradicts the fact that δd(G)=p2.Hence G is a 2-group of maximal class.Moreover,GQ8by Theorem 3.1.

To classify all finite p-groups with δd(G)=p,we need the following results.

Proposition 4.1LetGbe afinitep-group with|G|=pn.Then0＜ δk(G)＜ p2for somek ∈ {2,···,d}if and only ifk=dandδd(G)=p.

ProofThe necessity is trivial.

Conversely,suppose that 0 ＜ δk(G)＜ p2for some k ∈ {2,···,d}.It follows from Lemma 2.1 that k=d.Notice that 0＜δd(G)＜p2.Then G is not elementary abelian and so 1＜d＜n.If G is a 2-group of maximal class,then δd(G)=0 or 22by Theorem 3.1 and Theorem 4.1.This contradicts the fact that 0＜ δd(G)＜ p2.Hence G is neither cyclic nor a 2-group of maximal class.It follows from Lemma 2.3 that sn−d(G)= δd(G)+1 ≡ 1+p(mod p2).This together with 0 ＜ δd(G)＜ p2implies that δd(G)=p.

Lemma 4.1LetGbe afinitep-group with|G|=pn(n ≥ 3)andd≥ 2.Ifδd(G)=p,thenδd−1(G)=0.

ProofIt suffices to prove that each(d−1)-maximal subgroup H contains Φ(G).

If H is cyclic,then Φ(H)is a d-maximal subgroup of G.Hence|Φ(H)|=|Φ(G)|.Clearly,Φ(H)≤ Φ(G).Therefore Φ(G)= Φ(H)is a subgroup of H.

If H is not cyclic,then H has at least 1+p maximal subgroups by Lemma 2.2.But G has only 1+p d-maximal subgroups.Notice that every maximal subgroup of H is a d-maximal subgroup of G.Hence H has just 1+p maximal subgroups which are d-maximal subgroups of G.Thus the d-maximal subgroup Φ(G)of G must be a maximal subgroup of H.

Now we give the classification of finite p-groups G with δd(G)=p.

Theorem 4.2LetGbe afinitep-group of orderpnwithd ≥ 2.Thenδd(G)=pif and only ifGis one of the following non-isomorphic groups:

(1)Cpn−1×Cp(n ≥ 3);

(2)Mp(n−1,1),wheren≥4ifp=2;

(3)Q8×C2.

ProofIf G is one of the groups listed in the theorem,then it is easy to verify that δd(G)=p.

Conversely,suppose that G is a finite p-group of order pnsuch that δd(G)=p.Then δd−1(G)=0 by Lemma 4.1.Notice that G has exactly 1+p d-maximal subgroups.If there is no cyclic(d−1)-maximal subgroup of G,then each(d−1)-maximal subgroup can be generated by the 1+p d-maximal subgroups by Lemma 2.2.Thus G has only one(d−1)-maximal subgroup.This contradicts δd−1(G)=0.Therefore G has a cyclic(d − 1)-maximal subgroup.Moreover,the number of noncyclic(d−1)-maximal subgroups of G is at most 1.

If d≥3,then G is isomorphic to one of the groups listed in Theorem 3.2.By checking the groups listed in Theorem 3.2,it follows that GQ8×C2.Assume that d=2.Then G has a cyclic maximal subgroup.If G is abelian,then GCpn−1× Cp(n ≥ 3).If G is nonabelian,then G is isomorphic to one of the groups listed in Lemma 2.10.By checking the groups listed in Lemma 2.10,it follows that GMp(n−1,1),where n≥4 if p=2.

AcknowledgementsThe authors would like to express their gratitude to the referees for their valuable remarks and suggestions which improved the paper,and to Professor Qinhai Zhang for his helpful suggestions relating to this paper.

[1]An,L.J.and Liu,Y.D.,Characterization offinite p-group with the number of subgroups,Adv.Math.(China),40(3),2011,285–292.

[2]Berkovich,Y.,On the number of subgroups of given order in afinite p-group of expotent p,Proc.Amer.Math.Soc.,109,1990,875–879.

[3]Berkovich,Y.,Groups of Prime Power Order I,Walter de Gruyter,Berlin,New York,2008.

[4]Berkovich,Y.and Janko,Z.,Groups of Prime Power Order II,Walter de Gruyter,Berlin,New York,2008.

[5]Besche,H.U.,Eick,B.and O’Brien,E.A.,A millennium project:Constructing small groups,International J.of Algebra and Computation,12(5),2002,623–644.

[6]´Cepuli´c,V.,Ivankovi´c,M.and Kovaˇc Striko,E.,Second-metacyclicfinite 2-groups,Glas.Mat.Ser.,40(60),2005,59–69.

[7]Dyubyuk,P.E.,On the number of subgroups of certain categories offinite p-groups(in Russian),Mat.Sbornik N.S.,30(72),1952,575–580.

[8]Fan,Y.,A characterization of elementary abelian p-group by counting subgroups(in Chinese),Math.Pract.Theory,1,1988,63–64.

[9]Hall,P.,A contribution to the theory of groups of prime power order,Proc.London Math.Soc.,36,1933,29–95.

[10]Hua,L.K.,Some “Anzahl” theorems for groups of prime power orders,Sci.Rep.Nat.Tsing-Hua Univ.Ser.A,4,1947,313–327.

[11]Kulakoff,A.,über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in p-Gruppen,Math.Ann.,104,1931,778–793.

[12]O’Brien,E.A.and Vaughan-Lee,M.,The groups with order p7for odd prime p,J.Algebra,292,2005,243–258.

[13]Qu,H.P.,Finite non-elementary abelian p-groups whose number of subgroups is maximal,Israel J.Math.,195(2),2013,773–781.

[14]Qu,H.P.,Sun,Y.and Zhang,Q.H.,Finite p-groups in which the number of subgroups of possible order is less than or equal to p3,Chin.Ann.Math.Ser.B,31(4),2010,497–506.

[15]Tuan,H.F.,An “Anzahl” theorem of Kulakoff’s type for p-groups,Sci.Rep.Nat.Tsing-Hua Univ.Ser.A,5,1948,182–189.

[16]Zhang,Q.H.,A characterization of metacyclic p-groups by counting subgroups,Proc.of International Conference on Algebra 2010,World Sci.Publ.,Hackensack,NJ,2012,713–720.

[17]Zhang,Q.H.and Qu,H.P.,On Hua-Tuan’s conjecture,Sci.China Math.,52(2),2009,389–393.

[18]Zhang,Q.H.and Qu,H.P.,On Hua-Tuan’s conjecture II,Sci.China Math.,54(1),2011,65–74.

[19]Zhang,Q.H.,Zhao,L.B.,Li,M.M.and Shen,Y.Q.,Finite p-groups all of whose subgroups of index p3are abelian,Commun.Math.Stat.,3(1),2015,69–162.