Commutativity Criterions using Normal Subgroup Lattices

Commutativity Criterions Using Normal Subgroup Lattices

SIMIONBREAZ(*)

ABSTRACT- We prove that a groupGis Abelian whenever(1) it is nilpotent and the lattice of normal subgroups of Gis isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian groupBsuch that the normal subgroup lattice of BGis isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian groupAcan be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. ifAis an Abelian group andGis a group such thatZAandZGhave isomorphic normal subgroup lattices thenAandGare isomorphic groups.

1. Introduction.

IfGis a group, we will denote byL(G) the lattice of all subgroups ofG and byN(G) the lattice of all normal subgroups ofG. It is well known that there are many important properties of a group G which cannot be re- covered from properties ofL(G) orN(G). One of the simplest example is the following: G is a simple group if and only if N(G) is a chain with 2 elements, so if we have a groupGwith such a normal subgroup lattice we cannot say ifGis Abelian. However, there exist Abelian groupsAsuch that it is possible to deduce GA from the existence of an isomorphism N(G) N(A), by the works of Curzio, [4], and Brandl, [2]. Moreover, it is proved in [9] and [3] that subgroup lattices can be used to conclude thatGis Abelian: ifL(GG) orL(ZG) are modular lattices thenGis Abelian. In particular, ifL(GG) L(A) orL(ZG) L(A) forsome Abelian group

(*) Indirizzo dell'A.: BabesÎ-Bolyai University, Faculty of Mathematics and ComputerScience, Str. Mihail KogaÆlniceanu 1, 400084 Cluj-Napoca, Romania.

E-mail: [email protected]

1991Mathematics Subject Classification. 20E15; 20K27; 20F18.

The author is supported by the grant PN-II-CD ID-489

Athen Gis an Abelian group. A natural question is if we can replace in these results the subgroup lattice with the normal subgroup lattice.

First we observe that this transfer cannot be done in all cases.

Forexample, if G is a non-abelian simple group, then it is super- perfect (i.e. [N;G]ˆN forall N/G). Then [10, Theorem 1] shows that N(GG) f15Gg f15Gg  N(Z=2ZZ=3Z) (direct products of two chains with two elements).

In this paper we will prove that some criterions for the commutativity of a gr oupGcan be stated using normal subgroup lattices (Theorem 2 and Theorem 6): A groupGis Abelian if and only if one of the following con- ditions are satisfied:

(1)N(G) N(A) foran Abelian groupAandGis nilpotent;

(2)N(BG) N(A) forsome Abelian groupsBandAsuch thatBis not a torsion group.

In the end of the paper we will discuss cancellation properties which can be deduced from (2).

2. Direct products.

IfG_i,i2I, is a family of groups, we will denote by^5vQ

i2IG_itherestricted direct product of this family, i.e. the subgroup of all elements with finite support of the cartesian productQ

i2IG_i (see [8, pp.118±119]). If allG_i are Abelian then^5vQ

i2IG_iis denoted byL

i2IG_i, and it is called thedirect sumof the familyG_i,i2I. We note that, in order to simplify the exposition, we will identify every groupG_iwith the corresponding subgroup of^5vQ

i2IG_i. A well known theorem of Suzuki [12, Theorem 1.6.5] says us that the subgroup latticeL(G) of a groupGis a direct product of a family of lattices if and only if Gˆ^5vQ

i2IG_isuch that the groups G_i,i2I, ar e coprime, i.e.

everyG_iis a torsion group and gcd(o(g_i);o(g_j))ˆ1 foralli6ˆj,g_i2G_iand g_j 2G_j (here o(g) denotes the order of the element g). There exists a version of this theorem for normal subgroup lattices. It was proved for the caseIis finite by Millerin [10, Theorem 2]. The hypothesis ``Iis finite'' is not essential in Millers's proof. However, for reader's convenience, we include a proof of this theorem. In order to enunciate it, let us recall that a groupGissuperperfectif [N;G]ˆNforevery normal subgroupNofG.

THEOREM1. Let G be a group. The normal subgroup lattice of G has a direct decompositionN(G)ˆQ

i2IL_iif and onlyif Gˆ^5vQ

i2IG_isuch that (a)at most one of the groups Giis not superperfect

or

(b) for everyfamilyof normal subgroups H_i/G_i, the groups Hi=[Hi;Gi], i2I, are coprime.

In this caseN(Gi)ˆLi for all i2I.

PROOF. Suppose thatN(G)ˆQ

i2IL_i. Foreveryj2Iwe denote by 0_j and 1_j the least, respectively the greatest, element ofL_i. We denote by 1_jˆ(`<sub>i</sub>)<sub>i</sub>2 N(G) the element defined by conditions`_iˆ0_iforalli6ˆjand

`_j ˆ1_j. Foreveryj2Iwe considerthe normal subgroupsG_j ofGwhich correspond to 1j. Then foreveryj6ˆk2Ithe normal subgroupsGjandGk

centralize each other since G_j\G_kˆ f1g. Moreover, for every j2I we haveGˆGj S

i6ˆjGi

andGj\ S

i6ˆjGi

ˆ f1g, henceGˆ^5vQ

i2IGi. Moreover, sinceN(G)ˆQ

i2IN(Gi), it follows that every normal subgroupNofGis a direct product of normal subgroupsN_iof the groupsG_i.

Suppose that there exists j6ˆk in Isuch that G_j and G_k are not su- perperfect groups. Then there are normal subgroupsHj/Gj andHk/Gk

such that H_j 6ˆ[H_j;G_j] and H_k6ˆ[H_k;G_k]. If the Abelian groups H_j=[H_j;G_j] andH_k=[H_k;G_k] are not coprime then, by Suzuki's Theorem, there exists a subgroup

NH_j=[H_j;G_j]H_k=[H_k;G_k]

which is not a direct product of type K_jK_k withK_j H_j=[H_j;G_j] and K_k H_k=[H_k;G_k]. Then the subgroupNH_jH_k, which is determined by the properties [Hj;Gj][Hk;Gk]NandNˆN=([Hj;Gj][Hk;Gk]), is not a direct product of typeKjKk withKj Hj andKkHk, a con- tradiction.

Conversely, let N be a normal subgroup of G, and N_iˆp_i(N)/G_i, wherep_i:G!G_iare the canonical projections, for alli2I. Then

Y^5v

i2I

[N_i;G_i]NY^5v

i2I

N_i:

If we are under the hypothesis (a), so there existsj2Isuch that all groups

G_i(i6ˆj) are superperfect, thenNˆN_j^5vQ

i6ˆjN_i. If the hypothesis (b) is satisfied, then we can apply again Suzuki's Theorem to observe that N=^5vQ

i2I[N_i;G_i]ˆ^5vQ

i2IK_i=[N_i;G_i] for some normal subgroupsK_i/N_i, hence Nˆ^5vQ

i2IK_i, and the proof is complete. p

3. Commutativity criterions.

Using Theorem 1, we can prove a criterion for the commutativity of a nilpotent group.

THEOREM2. A nilpotent group G is Abelian if and onlyif there exists an abelian group A such thatN(G) N(A).

PROOF. LetW:N(G)! N(A) be an isomorphism.

Suppose thatGis not abelian. Then, by [12, Theorems 9.1.12, 9.1.14],A is a torsion group. We decomposeAas a direct sum of itsp-components:

AˆL

p A_p. ThenGˆ^5vQ

p W ¹(A_p), as a consequence of Theorem 1. It fol- lows that there exists a primepsuch thatGpˆW ¹(Ap) is not Abelian.

Therefore we can suppose thatAis ap-group. Using [12, Theorem 9.1.11], we observe thatAis locally cyclic, hence its subgroup lattice is a chain by [5, Theorem 3.1]. Therefore N(G)ˆ f15N₁5 5N_k5 g. Letc2N be the nilpotency class ofG. SinceGis not Abelian,c2. IfZi,i2 f0;. . .cg, denotes the terms of the upper central series ofG, then there existsksuch thatZ_{c 1}ˆN_k. Hence in the groupHˆG=Z_{c 2}, we have the properties:

Z(H)ˆZ_{c 1}=Z_{c 2}, and H=Z(H) is abelian. Moreover, every subgroup of H=Z(H) is of the form (N=Z_{c 2})=Z(H), with N normal in G. Hence the subgroup lattice of the Abelian groupH=Z(H) is a chain, henceH=Z(H) is a union of an ascending chain of cyclic subgroups (see [5, Section 3]).

Ifx;y2Hthen there existsu2Hsuch thatxZ(H);yZ(H)2 huZ(H)i.

Therefore xˆu^kv andyˆu<sup>`</sup>zforsome integersk and` and some ele- mentsv;z2Z(H). Then xyˆu^k‡`vzˆyx, henceHis abelian. It follows thatZc 1ˆG, a contradiction. ThereforeGis abelian. p EXAMPLE3. The previous result is not valid for solvable groups, since the latticesN(S₃) andN(Z=2²Z) are chains with 3 elements, hence they are isomorphic.

COROLLARY 4. If G is a finite p-group such that N(G) N(A) for some Abelian groupA, thenGis Abelian.

REMARK5. a) Corollary 4 does not work for subgroup lattices of finite p-groups, as a consequence of [12, Theorem 2.5.9].

b) Since there exists infinite simplep-groups, Corollary 4 is not valid for generalp-groups.

However, we can enunciate a result which is valid for all groups.

THEOREM6. Let B be an Abelian group which is not a torsion group. A group G is Abelian if and onlyif there exists an abelian group A such that N(BG) N(A).

PROOF. Let W:N(BG) N(A) be an isomorphism. Since every subgroup ofBis normal inBG, the restriction ofWtoBinduces a pro- jectivity fromBontoW(B) (i.e.W_jL(B) :L(B)! L(W(B)) is an isomorphism).

ThenL(W(B)) is not an atomic lattice (i.e. there exists an element which is not minimized by an atom), henceW(B) is not a torsion group. Therefore,A is not a torsion group.

If the torsion-free rank ofAis>1 thenABGas a consequence of [12, Theorem 9.1.12].

If A is of torsion-free rank 1, using the direct decomposition AˆW(B)W(G), we observe thatW(G) is a torsion group. For every ele- mentx2Aof infinite order we know thathxiand the torsion partT(A) of Aare disjoint (normal) subgroups ofA. HenceW ¹(hxi) andGare disjoint normal subgroups of BG, and it follows that G is a subgroup of the centralizer ofW ¹(hxi) forallx2AnT(A). Therefore,Gis contained in the centerofBG, since

BGˆW ¹(A)ˆW ¹(__x2AnT(A)hxi)ˆ __x2AnT(A)W ¹(hxi);

henceGis Abelian. p

EXAMPLE7. The hypothesis ``Bis not a torsion group'' is essential in the previous theorem.

PROOF. LetBˆZ=5ZandGˆS3. Using Theorem 1 we deduce that N(BG) N(B) N(G)ˆ f05Bg f15A₃5S₃g:

But this lattice is isomorphic to the subgroup lattice ofZ=5ZZ=2²Z. p

REMARK 8. As a consequence of [12, Theorem 9.1.11], the property stated in the previous theorem is valid if we assume that all involved groups arep-groups.

However, if onlyB6ˆ1 andGarep-groups the property is not valid. To see this, it is enough to considerBˆZ=pZ,Gan infinite simplep-group and AˆZ=pZZ=qZ (where q6ˆp are prime integers), and to apply Theorem 1.

It is proved in [3] that every abelian group is determined in the class of all groups by some subgroup lattices. As a consequence of Theorem 6, we obtain similarresults fornormal subgroup lattices.

COROLLARY9. LetGbe a group.

a) If B6ˆ1is a torsion-free Abelian group, and T is a torsion Abelian group such thatN(BT) N(BG), then TG.

b) If A is an Abelian group and B6ˆ1 is a finite rank torsion-free Abelian group such thatN(BA) N(BG), then there exists a posi- tive integer n such that AⁿGⁿ.

c) If A is an Abelian group andN(ZA) N(ZG)then AG.

d) If A is an Abelian group andN(QA) N(QG)then AG.

PROOF. We know thatGis Abelian by Theorem 6.

For a) we apply [12, Theorem 2.6.10 and Theorem 2.6.15].

Forb) we use a) and [12, Theorem 2.6.10] to deduce BABG.

Then we apply the ``power cancellation property'' proved by Goodearl in [6, Theorem 5.1].

The proofs for c) and d) are similar, using this time the fact thatZ and Q are cancellable from direct products of Abelian groups (see [1,

Corollary 8.8]). p

REMARK10. As a consequence of [13, Theorem 13], there exist non- isomorphic groupsGandHsuch thatZGZH. Then Corollary 9 c) cannot be extended to general groups. However, if ZGZH for some non-isomorphic groupsGandH, these groups must be infinite, as a consequence of [13, Theorem 11], so a natural question is if we can extend Corollary 9 to non-Abelian finite groups.

The answerforthis question is negative, and it is based on a (negative) solution which was found by Kearnes and Szendrei [7] for the problem for- mulated in [11]: ``can we deduceGHfromL(Gⁿ) L(Hⁿ) foralln1?''.

To present this answer, let us recall that two groups (G;) and (G;), defined on the same setG, ar eterm equivalentif they have the same term functions. Consequently, if (G;) and (G;) are term equivalent groups then

(T1) they have the same identity, and

(T2) the operationsandare term operations with respect toand, respectively.

We recall that aterm operationwith respect tois a function

GG!G; (x;y)7!T(x;y)ˆx^m1yⁿ¹ x^{m`</sup>y<sup>n`} forallx;y2G;

whereTˆX^m1Yⁿ¹ X^{m`</sup>Y<sup>n`} is abinaryterm(m_i;n_i2Zforalli).

The proof of the following lemma is included in the proof of [7, Lemma 2.4].

LEMMA11. Let(G;)be a group and T a binaryterm. If the pair(G;), where xyˆT(x;y), is a group with the same identityas(G;)then there exist r₁;. . .;r_k2Z, s₁;. . .;s_k 2Zande₁;. . .;e_k2 f1gsuch that

(])T(x;y)ˆxy[x^r1;y^s1]^e1. . .[x^rk;y^sk]^ek for all x;y2G (here[a;b]ˆaba ¹b ¹).

PROOF. There exist integers m1;. . .;m_k;n1;. . .;n_k such that we can write T(x;y)ˆxyx^m1yⁿ¹ x^mky^nk forall x;y2G. From T(x;1)ˆx, r e- spectivelyT(1;y)ˆy, forallx;y2G, we deduce

x^m1‡...‡mkˆy^n1‡...‡nk ˆ1:

Moreover, for allx;y2Gwe have T(x;y)ˆxyx^m1yⁿ¹ x^mky^nk

ˆxy[x^m1;yⁿ¹]yⁿ¹x^m1‡m2yⁿ² x^mky^nk ˆ

ˆxy[x^m1;yⁿ¹][x^m1‡m2;yⁿ¹] ¹x^m1‡m2y^n1‡n2 ;

and so on, hence the functionT(x;y) has the form (]). p REMARK12. IfGˆ(G;) andHˆ(G;) are groups as in Lemma 11, then foreveryx2G, its inversex ¹inGis the inverse ofxinH. Therefore, to prove that two groupsGˆ(G;) andHˆ(G;) are term equivalent, it is enough to prove that they satisfy the conditions (T1) and (T2).

COROLLARY 13. If Gˆ(G;) and Hˆ(G;) are term equivalent groups, thenZGandZH are term equivalent groups.

PROOF. By Lemma 11, there is a binary term TˆXY[X^r1;Y^s1]^e1 [X^rk;Y^sk]^ek

with r₁;. . .;r_k 2Z,s₁;. . .;s_k2Zande₁;. . .;e_k 2 f1gsuch that the op- erationis the term function induced byT.

If we considerthe term function induced byTonZGwe observe that T((m;x);(n;y))ˆ(m‡n;xy) forall m;n2Z and x;y2G. Then the operation onZHis a term operation on the setZGwith respect to the operation of the groupZG. By symmetry, the operation onZGis also a term operation with respect to the operation ofZH. It is not hard to see that the identities of the groups ZG andZHare the same.

Therefore, the groupsZGandZHare term equivalent. p COROLLARY 14. There exist non-isomorphic finite groups Gand H such that the latticesN(ZG)andN(ZH)are isomorphic.

PROOF. By [7, Example 2.14], there exist non-isomorphic term equivalent finite groups G and H. Then ZG and ZH are non-iso- morphic term equivalent groups. Applying [7, Lemma 2.6], we obtain

N(ZG)ˆ N(ZH). p

Acknowledgments. I would like to thank the referee for his/her careful reading of the paper and helpful critical remarks and suggestions.

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Manoscritto pervenuto in redazione il 28 novembre 2008.