1 Indecomposability and CD-minimal groups
Let G be a finite group and let H≤G. Then mG(H)=|H||CG(H)| is the Chermak–Delgado measure of H in G. Let m*(G)=max{mG(H):H≤G} and then define 𝒞𝒟(G)={H≤G:mG(H)=m*(G)}. The subgroup collection 𝒞𝒟(G) forms a sublattice of the lattice of subgroups of G. Furthermore, if H,K∈𝒞𝒟(G) and σ∈Aut(G), then we have Hσ∈𝒞𝒟(G), 〈H,K〉=HK, CG(H)∈𝒞𝒟(G), CG(H∩K)=CG(H)CG(K), and CG(CG(H))=H. The least element, T, of 𝒞𝒟(G) is known as the Chermak–Delgado subgroup ofG, and T is a characteristic, abelian subgroup of G which contains the center of G.
This modular, self-dual sublattice of the lattice of subgroups of G was first introduced in [7]. Proofs of the properties of 𝒞𝒟(G) stated in the previous paragraph are found in [10, Section 1.G].
We lay down some scaffolding that is useful in proving results about the Chermak–Delgado lattice. Proposition 1 below is adapted from [1, proof of Lemma 2.4.1].
Proposition 1.
Suppose H,K≤G, and HHx=HxH for all x∈K. If K≤HK, then K≤H.
Proof.
Given x,y∈K, HxHy, as x-conjugate of the subgroup HHyx-1, is a subgroup, and so HxHy=HyHx.
If H is normalized by K, the result is trivially true. So suppose there exists x∈K such that H≠Hx. By the initial remark, HHx satisfies the hypotheses of the proposition (instead of H). Therefore, by induction
on |HK:H|, we can assume that K≤HHx. Then x=h1h2x for h1,h2∈H which implies x∈H, a
contradiction.
∎
Corollary 1.
Suppose H,K≤G, and HHx=HxH for all x∈K. If H<K, then HK<K.
Proof.
If H<K, then HK≤K. Now H<K implies that K≰H so by the contrapositive of Proposition 1, K≰HK, and so in particular HK≠K. It follows that HK<K.
∎
If H∈𝒞𝒟(G), then any x∈G induces an automorphism of G, and thus Hx∈𝒞𝒟(G), and so 〈H,Hx〉=HHx, i.e. HHx=HxH. Thus Proposition 1 and Corollary 1 apply for any subgroup H∈𝒞𝒟(G) and any subgroup K of G.
Given H≤G and X⊆G, we use coreX(H) to denote intersection of all of the conjugates of H in X. We note that if H∈𝒞𝒟(G) and ∅≠X⊆G, then HX and coreX(H) are both in 𝒞𝒟(G).
Proposition 2.
The following statements hold:
If H∈𝒞𝒟(G) with H<G, then HG<G.
If K∈𝒞𝒟(G) with Z(G)<K, then Z(G)<coreG(K).
Proof.
Part (1) follows directly from Corollary 1.
If Z(G)<K, then CG(K)<G, so by Corollary 1, CG(K)G<G. And so Z(G)<CG(CG(K)G)=coreG(CG(CG(K))=coreG(K).
∎
Given H,K∈𝒞𝒟(G), we use the notation H≺K to mean that H<K and there is no R∈𝒞𝒟(G) so that H<R<K. If M is the greatest element in 𝒞𝒟(G) and T is the least element in 𝒞𝒟(G) (i.e. T is the Chermak–Delgado subgroup of G), we say that A∈𝒞𝒟(G) is an atom if T≺A, and we say that B∈𝒞𝒟(G) is a coatom if B≺M. Proposition 3 below appears in [6]. The proof in [6] is different than our approach.
Proposition 3.
Let H,K∈CD(G) with H≺K. Then H⊴K. And so maximal chains in CD(G) form subnormal series of G.
Proof.
We have
H<K and so by Corollary 1, HK<K, and we know that HK∈𝒞𝒟(G). Since H≺K, it must be that HK=H, i.e. H⊴K.
Now the greatest element in 𝒞𝒟(G) is normal in G (in fact characteristic in G), and so maximal chains in 𝒞𝒟(G) form subnormal series of G.
∎
Corollary 2.
Suppose G∈CD(G). Then all of the atoms and all of the coatoms of CD(G) are normal in G.
Proof.
If B∈𝒞𝒟(G) is a coatom, then B≺G, and so by Proposition 3, B⊴G. If A∈𝒞𝒟(G) an atom, then CG(A)∈𝒞𝒟(G) is a coatom, and so CG(A)⊴G, and so CG(CG(A))=A⊴G.
∎
The topic of this paper are those groups whose Chermak–Delgado subgroup is the identity subgroup. Consider direct products. Given H≤G1×…×Gn=G, one sees that CG(H)=CG(π1(H)×…×πn(H)), where πi is the projection map into the i-th coordinate. From here one sees that the Chermak–Delgado lattice of a direct product is the Cartesian product of the Chermak–Delgado lattices. Proposition 4 appears in [6]:
Proposition 4.
We have CD(G1×…×Gn)=CD(G1)×…×CD(Gn).
Corollary 3.
We have 1∈CD(G1×…×Gn) if and only if 1∈CD(Gi) for each i.
Proposition 5.
Suppose 1∈CD(G). If H,K∈CD(G) so that H∩K=1 and G=HK, then G=H×K is a direct product.
Proof.
Suppose H,K∈𝒞𝒟(G) so that H∩K=1 and G=HK. Let A=coreK(Z(H)) and let B=coreA(K). Note that A and B normalize one another, and A∩B=1, and thus A=coreK(Z(H))≤CG(coreA(K))=CG(K)A. So by Proposition 1, A≤CG(K). And since A≤Z(H) and G=HK, we have A≤Z(G)=1. It follows that 1=coreK(Z(H))=coreG(Z(H)), and so by Proposition 2 (2), Z(H)=1. Now H∈𝒞𝒟(G) and m*(G)=|G|. Hence
|G|=|H||CG(H)|=|H||CG(H)||Z(H)|=|HCG(H)|,
and so G=HCG(H), and so H⊴G. By a similar argument, switching H with K, one obtains that K⊴G. And so G=H×K is a direct product.
∎
How essential is it in Proposition 5 that 1∈𝒞𝒟(G)? Can one prove that if H,K∈𝒞𝒟(G) with H∩K=Z(G) and G=HK, then H⊴G and K⊴G? The answer is no, and so the property that 1∈𝒞𝒟(G) is needed. We quote [6, Example 1.7]. Let
G=〈a,b,c,d:a4=b2=c2=d2=[a,b]=[b,c]=[b,d]=[c,d]=[a,c]b=[a,d]c=1〉
Then |G|=32, Z(G)=〈b〉 has order 2, and m*(G)=64. Let H=〈a,b〉 and K=〈b,da〉. One can verify that H,K∈𝒞𝒟(G), H∩K=Z(G), G=HK, but neither H nor K is normal in G.
Corollary 4.
Suppose 1∈CD(G). If G=AB, where A and B are abelian subgroups of G, then G=1.
Proof.
Since A and B are abelian, A∩B≤Z(G)=1. So,
|A||B|=|G|=mG(G)≥mG(A)=|A||CG(A)|≥|A||A|,
so |B|≥|A|. Similarly,
|A||B|=|G|=mG(G)≥mG(B)=|B||CG(B)|≥|B||B|,
so |A|≥|B|; and it follows that |A|=|B|, and we have equalities everywhere in the previous two strings of inequalities. So A,B∈𝒞𝒟(G) and so by Proposition 5, G=A×B, and so G is abelian, and so 1=Z(G)=G.
∎
Theorem 1.
Suppose 1∈CD(G), and suppose CD(G)≅L1×L2 for lattices L1 and L2. Then G=H×K for subgroups H and K with CD(H)≅L1 and CD(K)≅L2.
Proof.
Suppose 1∈𝒞𝒟(G) and suppose that 𝒞𝒟(G) is lattice isomorphic to a Cartesian product ℒ1×ℒ2 of lattices ℒ1 and ℒ2. Let M1 and B1 be the greatest and least elements, respectively, of ℒ1 and let M2 and B2 be the greatest and least elements, respectively, of ℒ2. Note that (B1,M2)∨(M1,B2)=(M1,M2) is the greatest element of ℒ1×ℒ2 and (B1,M2)∧(M1,B2)=(B1,B2) is the least element of ℒ1×ℒ2. And since 𝒞𝒟(G) is lattice isomorphic to ℒ1×ℒ2, there is H,K∈𝒞𝒟(G) corresponding to (M1,B2),(B1,M2), respectively, in ℒ1×ℒ2 so that H∩K=1 and HK=G. By Proposition 5, G=H×K is a direct product. By Proposition 4, 𝒞𝒟(G)=𝒞𝒟(H)×𝒞𝒟(K). And so we have a lattice isomorphism between 𝒞𝒟(H)×𝒞𝒟(K) and ℒ1×ℒ2, where H corresponds to (M1,B2) and K corresponds to (B1,M2). Thus 𝒞𝒟(H)≅ℒ1 and 𝒞𝒟(K)≅ℒ2.
∎
By induction, this theorem extends to a direct product/Cartesian product of n number of groups/CD lattices.
A group G is said to be indecomposable if G cannot be written as an (internal) direct product H×K with H≠1 and K≠1. A lattice ℒ is said to be indecomposable if ℒ is not lattice isomorphic to a Cartesian product ℒ1×ℒ2 of lattices with ℒ1 and ℒ2 both nontrivial (a trivial lattice is a lattice consisting of a single point).
Corollary 5.
Suppose 1∈CD(G). Then G is indecomposable if and only if CD(G) is indecomposable.
Proof.
Suppose 1∈𝒞𝒟(G) and suppose that 𝒞𝒟(G) is indecomposable. If G=H×K, then by Proposition 4, 𝒞𝒟(G)=𝒞𝒟(H)×𝒞𝒟(K). Since 𝒞𝒟(G) is indecomposable, at least one of 𝒞𝒟(H) or 𝒞𝒟(K) is trivial, say without loss of generality that 𝒞𝒟(H) is trivial. By Corollary 3, since 1∈𝒞𝒟(G), 1∈𝒞𝒟(H), and so 𝒞𝒟(H)={1}. And so H=1, and thus G is indecomposable.
Suppose 1∈𝒞𝒟(G) and suppose G is indecomposable. Suppose that 𝒞𝒟(G) is lattice isomorphic to a Cartesian product ℒ1×ℒ2 of lattices ℒ1 and ℒ2. By Theorem 1, we have G=H×K for subgroups H and K with 𝒞𝒟(H)≅ℒ1 and 𝒞𝒟(K)≅ℒ2. And since G is indecomposable, at least one of H or K is trivial, and so at least one of ℒ1 or ℒ2 is trivial, and so 𝒞𝒟(G) is indecomposable.
∎
In [11], the author studied CD-simple groups, which are groups, G, having the property that 𝒞𝒟(G)={1,G}. We define a group, G, to be CD-minimal if 1∈𝒞𝒟(G) and G is indecomposable. So every CD-simple group is CD-minimal, but not vice versa.
Proposition 6.
Suppose G is CD-minimal but not CD-simple. If A∈CD(G) is an atom and if B∈CD(G) is a coatom, then A≤B.
Proof.
By Corollary 2, both 1≠A and 1≠B are normal in G. If A∩B=1, then G=A×B is a direct product, contrary to the assumption that G is indecomposable. So A∩B≠1, and since A is an atom in 𝒞𝒟(G), A≤B.
∎
Proposition 7.
Suppose G is CD-minimal but not CD-simple. If A∈CD(G) is an atom, then A is abelian, A⊴G, and |A| contains primes p≠q.
Proof.
Let 1≠A∈𝒞𝒟(G) be an atom. Then CG(A)∈𝒞𝒟(G) is a coatom, and by Proposition 6, A≤CG(A), i.e. A is abelian. By Corollary 2, A⊴G. Let N≤A be a minimal normal subgroup of G. So |N|=pk for some prime p and some k. Now, as N is minimal normal, G/CG(N) acts faithfully and irreducibly on N, and so |G/CG(N)| is not a power of p. And so a prime q≠p divides |G/CG(N)|. Now |G/CG(N)| divides |G/CG(A)|, and since A∈𝒞𝒟(G), we have |A|=|G/CG(A)|. Thus |A| contains primes p≠q.
∎
Lemma 1.
If H,K∈CD(G) and K≤H, then |H:K|=|CG(K):CG(H)|.
Proof.
As H,K∈𝒞𝒟(G), it follows that |H||CG(H)|=|K||CG(K)|, and so |H:K|=|CG(K):CG(H)|.
∎
Corollary 6 follows immediately:
Corollary 6.
Suppose G is CD-minimal but not CD-simple. If B∈CD(G) is a coatom, then CG(B)≤B, B⊴G, and |G:B| contains primes p≠q
Corollary 7.
Suppose 1∈CD(G). If 1≠H∈CD(G), then H is not a p-group for any prime p.
Proof.
If we prove Corollary 7 for any CD-minimal group, then by Corollary 3, the result will follow for any arbitrary group X with 1∈𝒞𝒟(X). So suppose G is a CD-minimal group. If G is CD-simple, then G is not a p-group for any prime p since Z(G)=1. If G is CD-minimal and not CD-simple, and 1≠H∈𝒞𝒟(G), then H contains an atom A∈𝒞𝒟(G), and so by Proposition 7, H is not a p-group for any prime p.
∎
Corollary 8.
Suppose 1∈CD(G). If G≠H∈CD(G), then |G:H| contains at least two different primes, and it follows that H is not a maximal subgroup of G.
Proof.
If we prove Corollary 8 for any CD-minimal group, then by Corollary 3, the result will follow for any arbitrary group X with 1∈𝒞𝒟(X). So suppose G is a CD-minimal group. If G is CD-simple, then 1=Z(G)=H≠G implies that |G|=|G:1| contains at least two different primes. So suppose that G is CD-minimal and not CD-simple. Suppose G≠H∈𝒞𝒟(G). Then we have H≤B, where B∈𝒞𝒟(G) is a coatom. Note that |G:B| divides |G:H|, and by Corollary 6, |G:B| contains primes p≠q, and so |G:H| contains primes p≠q. Suppose by way of contradiction that H is a maximal subgroup of G and let P be a Sylow p-subgroup of G. Then H<HP. Since H is maximal in G, HP=G. But |G:HP| contains a prime q, a contradiction.
∎
Denote the class of all finite groups with a trivial Chermak–Delgado subgroup by 𝒯. The reader summarizes results as they pertain to 𝒯. The class 𝒯 non-trivially intersects the class of solvable groups, as the symmetric group S4 is CD-simple. And the class 𝒯 properly contains the class of non-abelian simple groups. All of the examples of CD-minimal groups in the next section are solvable groups.
Given G∈𝒯, 𝒞𝒟(G) does not contain any non-trivial p-groups by Corollary 7. It was shown in [11] (see Proposition 6 there) that given G∈𝒯, 𝒞𝒟(G) does not contain any non-trivial, cyclic subgroups that are normal in G. We can apply Proposition 2 (2) to show that given G∈𝒯, 𝒞𝒟(G) does not contain any non-trivial, cyclic groups.
Let 𝒯* denote the class of all lattices 𝒞𝒟(G) with G∈𝒯. The class 𝒯* is closed under Cartesian products by Corollary 4, and if ℒ∈𝒯* is lattice isomorphic to a Cartesian product of lattices, then each lattice appearing in that Cartesian product is in 𝒯*; this is by Theorem 1. Given indecomposable ℒ∈𝒯* with |ℒ|>2, and given X∈ℒ an atom, and Y∈ℒ a coatom, we have that X≤Y by Proposition 6. A lattice ℒ is said to be a quasi-antichain of widthn if ℒ contains exactly n atoms, and every atom in ℒ is a coatom in ℒ. We denote a quasi-antichain lattice of width n by ℳn. And so given indecomposable ℒ∈𝒯*, ℒ is not isomorphic to ℳn for any n>1. Note that the group 1=G∈𝒯 has that 𝒞𝒟(G)={1}≅ℳ0, and in the next section we construct a group G∈𝒯 so that 𝒞𝒟(G)≅ℳ1, but beyond that, 𝒞𝒟(G) is never a quasi-antichain for indecomposable G∈𝒯. This is surprising considering the prevalence of ℳn lattices in the theory of Chermak–Delgado p-group lattices, see [5] and [2]. We will see in the next section that quasi-antichain p-group lattices do play a major role in the CD lattices of some CD-minimal group examples.
Suppose that ℒ is a lattice with greatest element M and least element B. Given X∈ℒ, we say that Y∈ℒ is a complement ofX if X∨Y=M and X∧Y=B. Given indecomposable ℒ∈𝒯*, it follows from Proposition 5 that the only elements in ℒ that have complements in ℒ are the greatest and least elements in ℒ. This further restricts the structure of indecomposable ℒ∈𝒯*.
2 Examples of CD-Minimal Groups which are not CD-Simple
Do there exist CD-minimal groups which are not CD-simple? The answer is yes!
In this section,
ℤn denotes the group of integers modulo n, Cn denotes the cyclic group of order n, Sn denotes the symmetric group on n elements, Q8 denotes the quaternion group of order 8, and QD16 denotes the quasidihedral group of order 16. Note that QD16=〈r,s:r8=s2=1,srs=r3〉. Given a group action of a group T on a group N, we let [N]T denote the semidirect product.
Lemma 2.
Suppose 1≠P is a p-group and T is a group so that T acts on P and so that the restricted action of T on Z(P) is faithful and irreducible, and let G=[P]T. If 1≠A⊴G and A is abelian, then Z(P)≤A≤CG(A)≤P.
Proof.
Note that if A∩Z(P)=1, then since A⊴G, and Z(P)⊴G, we have A≤CG(Z(P))=P, since T acts faithfully on Z(P). But since 1≠A⊴P and 1≠P is a p-group, we have A∩Z(P)>1, a contradiction. So A∩Z(P)>1. Now, since T acts irreducibly on Z(P), it follows that Z(P)≤A. And so we have A≤CG(A)≤CG(Z(P))=P.
∎
For i∈{1,2}, let Pi be a 2-group so that
there exists a group Ti=[Hi]Ki with |Hi|=3 and |Ki|=2 so that Ti acts on Pi and so that the restricted action of Ti on Z(Pi) is faithful and irreducible (and so Ti≅Aut(ℤ2×ℤ2)≅S3).
Let P3 be a 3-group so that P3∈𝒞𝒟(P3), so that Z(P3)≅ℤ3×ℤ3, and so that there exists a group T3=[H3]K3 with |H3|=8 and |K3|=2 so that T3 acts on P3 and so that the restricted action of T3 on Z(P3) is faithful and irreducible (note that a Sylow 2-subgroup of Aut(ℤ3×ℤ3) is isomorphic to QD16=[〈r〉]〈s〉.)
Let G1=[P1]T1, G2=[P2]T2, G3=[P3]T3, and let G=G1×G2×G3.
Let πi denote the projection homomorphism of G onto the ith coordinate. Let S≤G so that for each i∈{1,2,3}, one has πi(S)=Gi, S∩Gi=[Pi]Hi, and S∩(K1×K2×K3) is the diagonal subgroup of K1×K2×K3 (and so |S∩(K1×K2×K3)|=2.)
Lemma 3.
Let S be a group constructed as above. If H∈CD(S), and if for each i∈{1,2,3},
Z(Pi)≤πi(H)≤Pi,
then H,CS(H)∈(CD(P1)×CD(P2)×CD(P3)).
Proof.
Note that
CS(H)=CS(π1(H)×π2(H)×π3(H))≤CS(Z(P1)×Z(P2)×Z(P3))=P1×P2×P3
since each Ti acts faithfully on Z(Pi). And so
CS(H)=CP1(π1(H))×CP2(π2(H))×CP3(π3(H)).
And since H∈𝒞𝒟(S), it follows that H=π1(H)×π2(H)×π3(H). And so H,CS(H)∈(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3)).
∎
Theorem 2 below is an example of a CD lattice extension theorem. There have been papers written about CD lattice extension theorems for p-groups, see [4, 3]. This is the first non p-group CD lattice extension theorem that the author is aware of.
Theorem 2.
Let S be a group constructed as above, and suppose |P1|=2a, |P2|=2b, and |P3|=3c for some a,b,c. Then S is a CD-minimal group of order 2a+b+4⋅3c+2 and CD(S)=(CD(P1)×CD(P2)×CD(P3))∪{1,S}.
Proof.
Let S be a group constructed as above, and suppose |P1|=2a, |P2|=2b, and |P3|=3c for some a,b,c. And so we have |Z(P1)|=4, |Z(P2)|=4, and |Z(P3)|=9. And since P1∈𝒞𝒟(P1), P2∈𝒞𝒟(P2), and P3∈𝒞𝒟(P3), we have that
2a+b+4⋅3c+2=m*(P1)⋅m*(P2)⋅m*(P3)=|S|=|P1||H1||P2||H2||P3||H3|2.
We now establish that the Chermak–Delgado subgroup of S is 1, and along the way we determine all of the abelian, normal subgroups of S that are in 𝒞𝒟(S). Note that mS(1)=|S|. We know that the Chermak–Delgado subgroup of S is a characteristic, abelian subgroup of S, and so let us suppose that 1≠A⊴S with A abelian and A∈𝒞𝒟(S). Since A⊴S, for each i∈{1,2,3}, it follows that πi(A)⊴πi(S)=Gi. By Lemma 2, if πi(A)≠1, then
Z(Pi)≤πi(A)≤CGi(πi(A))≤Pi.
And so CGi(πi(A))=πi(CS(A)) and |πi(A)||πi(CS(A))|≤m*(Pi). We now show that for each i∈{1,2,3}, πi(A)≠1. Suppose not. Then there is j0∈{1,2,3} so that πj0(A)=1. And since 1≠A, there is i0∈{1,2,3} so that πi0(A)≠1. So if πj(A)=1, then πj(CS(A))=[Pj]Hj. This is true because Ti0 acts faithfully on Z(Pi0)≤πi0(A), and so the diagonal subgroup of K1×K2×K3 will not centralize A. Thus, if πj(A)=1, then
|πj(CS(A))|=|Pj||Hj|<|Pj||Z(Pj)|=m*(Pj).
And so
mS(A)=|A||CS(A)|≤|π1(A)||π2(A)||π3(A)||π1(CS(A))||π2(CS(A))||π3(CS(A))|<m*(P1)⋅m*(P2)⋅m*(P3)=|S|
which contradicts A∈𝒞𝒟(S).
So for each i∈{1,2,3}, πi(A)≠1. By Lemma 2, Z(Pi)≤πi(A)≤Pi for each i∈{1,2,3}. So by Lemma 3, A,CS(A)∈(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3)). Thus mS(A)=m*(P1)⋅m*(P2)⋅m*(P3)=|S|. Hence we have shown that the Chermak–Delgado subgroup of S is the identity, and hence m*(S)=|S|, and so
(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3))∪{1,S}⊆𝒞𝒟(S).
Furthermore, we know that if 1≠A⊴S with A abelian and A∈𝒞𝒟(S), then A∈(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3)).
Note that Z(P1)×Z(P2)×Z(P3) is an atom in 𝒞𝒟(S). This is true because otherwise, Z(P1)×Z(P2)×Z(P3) would properly contain an atom 1≠A of 𝒞𝒟(S). By Corollary 2, A⊴S. But then A∈(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3)), a contradiction. We now show that Z(P1)×Z(P2)×Z(P3) is the unique atom in 𝒞𝒟(S). Suppose not. So 1≠N is another atom in 𝒞𝒟(S). So by Corollary 2, we have N⊴S. Since N∩(Z(P1)×Z(P2)×Z(P3))=1, it follows that N≤CS(Z(P1)×Z(P2)×Z(P3))=P1×P2×P3. And since P1×P2×P3 is nilpotent, we have that N∩(Z(P1)×Z(P2)×Z(P3))>1, a contradiction. So Z(P1)×Z(P2)×Z(P3) is the unique atom in 𝒞𝒟(S).
If H∈𝒞𝒟(S) with H≠1 and H≠S, then
Z(P1)×Z(P2)×Z(P3)≤H,CS(H),
and so
CS(H),H≤P1×P2×P3,
and so by Lemma 3, H∈(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3)). Thus we have that 𝒞𝒟(S)=(𝒞𝒟(P1)×𝒞𝒟(P2)×𝒞𝒟(P3))∪{1,S}.
Finally, since 𝒞𝒟(S) contains a unique atom, it follows that 𝒞𝒟(S) is an indecomposable lattice. And so by Corollary 5, S is indecomposable. Thus, S is a CD-minimal group with the prescribed properties.
∎
We desire examples of groups P1,P2,P3 and acting groups T1,T2,T3 in Theorem 2. One can take P1=ℤ2×ℤ2, P2=ℤ2×ℤ2, P3=ℤ3×ℤ3, and T1=Aut(ℤ2×ℤ2), T2=Aut(ℤ2×ℤ2), and T3 to be a Sylow 2-subgroup of Aut(ℤ3×ℤ3). Let S0 be the CD-minimal group constructed using each of these examples. The author thanks Peter Hauck for providing this construction. The group S0 has order 28⋅34=20736. We have 𝒞𝒟(S0)≅ℳ1. Note that the unique atom/coatom, A=(ℤ2×ℤ2)×(ℤ2×ℤ2)×(ℤ3×ℤ3), in 𝒞𝒟(S0) is abelian, A⊴S0, and |A| contains at least two primes. This (thankfully) agrees with the theory established in Section 1.
Proposition 8 below appears in [5]. The proof makes use of [9, Exercise 39].
Proposition 8.
Let p be a prime and n a positive integer. Let P be the group of all 3×3 lower triangular matrices over GF(pn) with ones along the diagonal. The Chermak–Delgado lattice of P is a quasi-antichain of width pn+1 and all subgroups in the middle antichain are abelian.
Proposition 9.
Let p be a prime and n a positive integer. Let P be the group of all 3×3 lower triangular matrices over GF(pn) with ones along the diagonal. Then there exists T≤Aut(P) so that T=[H]K with |H|=pn-1 and |K|=n, and so that the restricted action of T on Z(P)≅Zp×…×Zpn times is faithful and irreducible.
Proof.
Note that
[100a110b1c11]⋅[100a210b2c21]=[100a1+a210b1+b2+c1a2c1+c21],
and note that
Z(P)={[100010b01]:b∈GF(pn)}≅ℤp×…×ℤpn times.
Let x be a generator of the group of units of GF(pn). Define
([100a10bc1])r=[100x⋅a10x⋅bc1].
Note that r∈Aut(P) and H=〈r〉 has order pn-1. Define
([100a10bc1])s=[100ap10bpcp1].
Note that s∈Aut(P) and K=〈s〉 has order n, and observe that s-1rs=rp, and T=[H]K≤Aut(P) and acts faithfully and irreducibly on Z(P).
∎
Proposition 9 with p=2 and n=2 yields examples of groups P1,P2 and T1,T2 in Theorem 2. Proposition 9 with p=3 and n=2 yields examples of groups P3 and T3 in Theorem 2. Proposition 8 tells us that each Pi∈𝒞𝒟(Pi) and tells us the structure of 𝒞𝒟(Pi).
The following example was provided by Ben Brewster, and yields examples of groups P1,P2 and T1, T2 in Theorem 2. Let D={(x,x,x):x∈Z(Q8)}, and let P=(Q8×Q8×Q8)/D. Consider the natural action of T=S3 on P via permutation of the coordinates. Then T acts faithfully and irreducibly on Z(P)≅ℤ2×ℤ2. It can be shown that
𝒞𝒟(P)={(X×Y×Z)/D:X,Y,Z∈𝒞𝒟(Q8)}.
The author verified this through use of GAP. We know that 𝒞𝒟(Q8) consists of all of the subgroups of Q8 that contain Z(Q8), and so 𝒞𝒟(Q8) is a quasi-antichain of width 3. And so 𝒞𝒟(P)≅ℳ3×ℳ3×ℳ3. The author would like to remark that Q8 is an example of an extraspecial p-group, and it is true that for any extraspecial p-group, R, 𝒞𝒟(R) consists of all of the subgroups of R that contain Z(R). This was mentioned in [8, Example 2.8]; the result follows from considering a non-degenerate bilinear form induced by commutation that is endowed upon R/Z(R) when viewed as a vector space over 𝔽p. The author would like to remark that in a recent paper, [12], the groups G having the property that 𝒞𝒟(G) consists of all of the subgroups of G that contain Z(G) are classified.
We encourage the reader to collect all of the above examples that work in Theorem 2. We arrive at twelve examples of CD-minimal groups that are not CD-simple, with each example having a different CD lattice. These lattices are extensions of Cartesian products of quasi-antichain lattices by a top and bottom element.
We end with some open questions. Are there examples of CD-minimal groups, not CD-simple, that are {p,q}-groups for {p,q}≠{2,3}? Obviously, one desires more theory on the class of groups 𝒯 and the class of lattices 𝒯*, and more examples of groups and lattices in each class would further that theory. Is the appearance of quasi-antichain p-group lattices in this construction mere coincidence, perhaps because of the small order of groups involved in the construction, or is there a deeper reason that quasi-antichain lattices appear prevalently in CD lattices? For groups in 𝒯 and lattices in 𝒯*, we have established a relationship between direct products of groups and Cartesian products of lattices. Are there more relationships between lattice theoretic properties of 𝒞𝒟(G) and group theoretic properties of G?
