Groups whose Chermak–Delgado lattice is a quasi-antichain

Theorem 1.1 ([1]).

If L is a Chermak–Delgado lattice of a finite p-group G such that both G/Z⁢(G) and G′ are elementary abelian, then so are L+ and L++, where L+ is a mixed 3-string with center component isomorphic to L and the remaining components being m-diamonds (a lattice with subgroups in the configuration of an m-dimensional cube), L++ is a mixed 3-string with center component isomorphic to L and the remaining components being lattice isomorphic to Mp+1 (a quasi-antichain of width p+1; see the following definition).

Theorem 1.2 ([3]).

If a finite group G∈C⁢D⁢(G) has a quasi-antichain of width w≥3, then

1. the group G is nilpotent of class 2 , both G/Z⁢(G) and G′ are elementary abelian p-groups for some prime p, and G/Z⁢(G) is a direct product of any pair of atoms modulo Z⁢(G);

2. there exists a non-abelian Sylow p-subgroup P and an abelian Hall p′-subgroup Q such that G=P×Q, P∈𝒞⁢𝒟⁢(G) and 𝒞⁢𝒟⁢(P)≅𝒞⁢𝒟⁢(G) as lattices;

3. the number t of abelian atoms is of the form t=pb+1 for some integer b (where t≥2).

Main theorem.

Suppose that G∈C⁢D⁢(G)≅Mw, and there are, in total, t>2 abelian atoms in C⁢D⁢(G). Then there are positive integers a and b and some prime p such that w=pa+1 and t=pb+1, where a=b or a=2⁢b. If

then |⁡an.

Proof.

By Theorem 1.2 (2), we may assume that G is a finite p-group.

Let M1,M2,…,Mw be all the atoms of 𝒞⁢𝒟⁢(G), M1=〈x1,x2,…,xn〉⁢Z⁢(G) and M2=〈y1,y2,…,yn〉⁢Z⁢(G) such that M1/Z⁢(G) and M2/Z⁢(G) are elementary abelian groups of order pn for some positive integer n. Then, by Theorem 1.2 (1), for k≥3, Mk=〈x1⁢y1′,x2⁢y2′,…,xn⁢yn′〉⁢Z⁢(G), where

Moreover, there is an invertible matrix Ck=(ci⁢j(k))n×n over Fp such that

In this paper, the matrix Ck is called the characteristic matrix of Mk relative to (x1,x2,…,xn) and (y1,y2,…,yn).

Without loss of generality, we may assume that M1, M2 and M3 are abelian atoms. For convenience, the operation of the group G/Z⁢(G) is written as addition. By Theorem 1.2 (1), M3=〈x1+y1′,x2+y2′,…,xn+yn′〉⁢Z⁢(G) such that

C3=(ci⁢j(3))n×n is the characteristic matrix of M3 relative to (x1,x2,…,xn) and (y1,y2,…,yn). Replacing y1,y2,…,yn with y1′,y2′,…,yn′ respectively, we may assume that C3=In.

Let zi⁢j=[xi,yj] and Z=(zi⁢j)n×n. Then G′=〈zi⁢j∣i,j=1,2,…,n〉. We have the following formalized calculation (where CT is denoted to the transpose of C):

Hence CG⁢(Mk)=Mk′ if and only if (Z⁢Ck)T=Z⁢Ck′. In particular, Mk is abelian if and only if Z⁢Ck is symmetric. Since M3 is abelian, ZT=Z.

Let 𝕍={C∈ℳn⁢(Fp)∣there exists⁢C′∈ℳn⁢(Fp)⁢such that⁢CT⁢Z=Z⁢C′}. Notice that if C∈𝕍 and CT⁢Z=Z⁢C′, then C′∈𝕍. It is straightforward to prove that 𝕍 is a linear space over Fp. Hence |𝕍|=pa for some positive integer a.

Assume that On≠C∈𝕍 and CT⁢Z=Z⁢C′. Let

Then, by the formalized calculation above, [(s1,s2,…,sn)T,(t1,t2,…,tn)]=On. Let

Then N2≤CG⁢(N1). Hence |N1|⁢|CG⁢(N1)|≥|N1|⁢|N2|=|M1|2=m*⁢(G). This implies that N1∈𝒞⁢𝒟⁢(G), and, there is some k≥3 such that N1=Mk. Hence 𝕍∖{On} is the set of Ck where 3≤k≤w. Since Ck is invertible, 𝕍 is a division algebra. By Wedderburn’s little theorem, 𝕍 is also a finite field. Now we know that w-2=pa-1. Notice that a≤n (see [3, Theorem 6]). Let A=(ai⁢j) be a primitive element of 𝕍, and mA⁢(λ)=k0+k1⁢λ+…+ka-1⁢λa-1-ka⁢λa the minimal polynomial of A. Then 𝕍={On,A,A2,…,Apa-1=In}=Fp(A).

Notice that M2/Z⁢(G) can be regarded as a n-dimensional vector space 𝕍2 over Fp, in which (y1⁢Z⁢(G),y2⁢Z⁢(G),…,yn⁢Z⁢(G)) is a basis for 𝕍2. We can make 𝕍2 an Fp⁢[λ]-module by defining the action of any polynomial g⁢(λ)=b0+b1⁢λ+…+bm⁢λm on any vector y∈𝕍2 as g⁢(λ)⁢y=b0+b1⁢(𝒜⁢y)+…+bm⁢(𝒜m⁢y), where 𝒜⁢(yj⁢Z⁢(G))=(∏i=1nai⁢j⁢yi)⁢Z⁢(G), j=1,2,…,n. By the well-known principal ideal domain theory and linear algebra arguments, 𝕍2 is a direct sum of r=n:a cyclic module. Thus

Suppose that AT⁢Z=Z⁢A1. Since A1∈𝕍, there is 1≤k≤pa-1 such that A1=Ak. Thus AT⁢Z=Z⁢Ak. Let

where Zi⁢j are a×a matrices. Since ZT=Z, Zj⁢i=Zi⁢jT for 1≤i≤j≤r. It follows from AT⁢Z=Z⁢Ak that

Let

Then

By (2.1) and (2.3),

Hence

That is, Zi⁢j(1)⁢mA⁢(Bk)=0. We claim that mA⁢(Bk)=0. Otherwise, mA⁢(Ak)≠0. Since mA⁢(Ak)∈𝕍 and 𝕍 is a field, mA⁢(Ak) is invertible. Hence mA⁢(Bk) is also invertible. Notice that if Zi⁢j(1)=0 then Zi⁢j=Oa by (2.4) and (2.5). Hence we may choose Zi⁢j such that Zi⁢j(1)≠0. In this case, Zi⁢j(1)⁢mA⁢(Bk)≠0, a contradiction. Thus mA⁢(Bk)=0 and mA⁢(Ak)=0. Since Ap,Ap2,…,Apa=A are all zero points of mA⁢(λ), there exists a 1≤e≤a such that k=pe. By easy calculation, (Am)T⁢Z=Z⁢Am⁢pe.

Let 𝕎={C∈𝕍∣CT⁢Z=Z⁢C}. Then |𝕎|=t-1 since t is the number of abelian atoms in 𝒞⁢𝒟⁢(G). Since Am⁢pe=Am if and only if |⁡(pa-1)m⁢(pe-1),

Hence 𝕎∖{On} is a cyclic group generated by Apa-1(pa-1,pe-1) of order

Let b=(a,e). Then t=pb+1. By (2.1),

By (2.2),

Hence |⁡(pa-1)(p2⁢e-1). Thus |⁡a2⁢e. It follows that |⁡a2⁢b. Hence a=e=b or a=2⁢e=2⁢b. ∎

Remark 3.1.

Let F be a field containing pa elements, and let F* be the multiply group of F. Then F* is cyclic with order pa-1. Let F*=〈b〉 and ℬ:f↦b⁢f be linear transformations over F, where F is regarded as a linear space over the field Fp. Of course, the order of ℬ is pa-1. Let p⁢(x) be the minimal polynomial of ℬ. It follows from the Cayley–Hamilton theorem that deg⁡(p⁢(x))=r≤a. Let W={f⁢(ℬ)∣f⁢(x)∈Fp⁢[x]}={f1⁢(ℬ)∣f1⁢(x)∈Fp⁢[x],deg⁡(f1)<r}. Then dim⁡W=deg⁡(p⁢(x))=r and hence |W|=pr. On the other hand,

and hence |W|≥pa. So we get r=a. Let the minimal polynomial of ℬ be

Then a matrix of ℬ is

Lemma 3.2.

Let B be the matrix introduced in Remark 3.1. Let Z=(zi⁢j) be an a×a matrix. If BT⁢Z=Z⁢B, then Z is symmetric.

Proof.

By calculation, the (u,v)-th entry of BT⁢Z is zu+1,v, while the (u,v)-th entry of ZB is zu,v+1, where 1≤u,v≤a-1. Hence zu+1,v=zu,v+1, where 1≤u,v≤a-1. Then, for u≤v, we have

Hence Z is symmetric. ∎

Theorem 3.3.

Suppose that a and r are positive integers. Let n=a⁢r, and let p be a prime. Then there exists a group G such that |G/Z⁢(G)|=p2⁢n, and C⁢D⁢(G) is a quasi-antichain of width w=pa+1, in which the number of abelian atoms is t=pa+1.

Proof.

We construct G=〈x1,x2,…,xn,y1,y2,…,yn〉 with defining relationships:

1. xip=yip=1, [xi,xj]=[yi,yj]=1, [xi,yj]=zi⁢j for all i and j such that 1≤i,j≤n;

2. z(u-1)⁢a+1,(v-1)⁢a+j∈Z⁢(G) for 1≤u≤v≤r and 1≤j≤a;

   (In the following, we use the addition operation to denote the multiplication operation.)

3. Zu⁢v(k)=Zu⁢v(1)⁢Bk-1 for 1≤u≤v≤r and 2≤k≤a, where

   Zu⁢v(k):=(z(u-1)⁢a+k,(v-1)⁢a+1,z(u-1)⁢a+k,(v-1)⁢a+2,…,z(u-1)⁢a+k,(v-1)⁢a+a);

4. Zu⁢v=Zv⁢uT for 1≤v<u≤r, where

   Zu⁢v=(z(u-1)⁢a+i,(v-1)⁢a+j)=(Zu⁢v(1)Zu⁢v(2)⋮Zu⁢v(a)).

It is easy to see that

is elementary abelian of order pr+12⁢n. Hence |G|=pr+52⁢n.

By relationship (3), BT⁢Zu⁢v=Zu⁢v⁢B for 1≤u≤v≤r. By Lemma 3.2, Zu⁢v is symmetric for 1≤u≤v≤r. Hence Zu⁢v=Zv⁢uT=Zv⁢u for 1≤v<u≤r. Moreover, Zu⁢vT=Zv⁢uT=Zu⁢v=Zv⁢u for all 1≤u,v≤r. Let

Then ZT=Z. Let X=〈x1,x2,…,xn〉⁢Z⁢(G) and Y=〈y1,y2,…,yn〉⁢Z⁢(G).

Assertion (1): CG⁢(x)=X for all x∈X∖Z⁢(G). Let x=∏i=1nci⁢xi+z, where z∈Z⁢(G). Write Ck=(c(k-1)⁢a+1,c(k-1)⁢a+2,…,ck⁢a) for 1≤k≤r. Since x∉Z⁢(G), there exists a k0 such that Ck0≠(0,0,…,0). Exchanging

we have Cr≠(0,0,…,0). Let Hi=〈[x,y(i-1)⁢a+j]∣1≤j≤a〉 for 1≤i≤r. We will prove that H1+H2+…+Hv is of order pv⁢a. Using induction, we may assume that H1+H2+…+Hv-1 is of order p(v-1)⁢a. By calculation,

Since Cr≠(0,0,…,0),

By Remark 3.1, ∑i=1ac(r-1)⁢a+i⁢Bi-1 is invertible. Hence

is of rank a. It follows that

Thus H1+H2+…+Hv is of order pv⁢a.

By the discussion above, H1+H2+…+Hr is of order pn, and hence |[x,G]|=pn. It follows that

Since |X|=pr+32⁢n and X≤CG⁢(x), CG⁢(x)=X.

Similarly, we have CG⁢(y)=Y for all y∈Y∖Z⁢(G). Then CG⁢(X)=X and CG⁢(Y)=Y, yielding mG⁢(G)=mG⁢(X)=mG⁢(Y)=p(r+3)⁢n.

Assertion (2): m*⁢(G)=p(r+3)⁢n and 𝒞⁢𝒟⁢(G) is a quasi-antichain. Otherwise, by the dual-property of 𝒞⁢𝒟-lattice, there exists H∈𝒞⁢𝒟⁢(G) such that

Since

there exists x∈H∩X∖Z⁢(G). Hence CG⁢(H)≤CG⁢(x)=X. Similarly, we have CG⁢(H)≤Y. Hence CG⁢(H)=Z⁢(G) and mG⁢(H)<mG⁢(G), a contradiction.

The discussion above also gives that 𝒞⁢𝒟⁢(G) is a quasi-antichain, in which every atom is of order p(r+3)2⁢n.

Assertion (3): w=t=1+pa. Now we have G,Z⁢(G),X,Y∈𝒞⁢𝒟⁢(G). Let M be an atom different from X and Y. Then, by the same reason given in the main theorem, we may let M=〈w1,w2,…,wn〉 and N=CG⁢(M)=〈v1,v2,…,vn〉, where

Since [M,N]=0, CT⁢Z=Z⁢D. Let