In particular, simple groups have no non-trivial set-direct factorization

DAN LEVY AND ATTILA MAR ´OTI

Abstract. We consider factorizationsG=XY whereGis a general group, XandY are normal subsets ofGand anyg∈Ghas a unique representation g=xy withx∈X andy∈Y. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a groupGhas such a factorization if and only if Gis a central product of ⟨X⟩and ⟨Y⟩and the central subgroup⟨X⟩ ∩ ⟨Y⟩satisfies certain abelian factorization conditions.

We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.

1. Introduction

Factorizations of groups is an important topic in group theory that has many facets. The most basic and best studied type of factorization is the direct product factorization of a group into two normal subgroups. IfGis a group andH andK are two normal subgroups of Gthen G=H ×K if and only if each g ∈Ghas a unique representationg=hk withh∈H andk∈K. One possibility to generalize this definition is to relax the condition that bothH andKare normal. This leads to the well-known concept of a semi-direct product of subgroups (only one of the factors is assumed to be normal) and also to the consideration of factorizations G =HK where neither of the two subgroupsH and K is normal, and even the unique representation condition is not necessarily assumed. To get a glimpse of the possibilities see the seminal classification result in [11] of maximal decompositions G=HKwhereGis a finite simple group andH andKare two maximal subgroups ofG.

Another generalization arose from a geometry problem of Minkowski [13]. In 1938 Haj´os [7] reformulated this problem as an equivalent factorization problem of a finite abelian group, where the factors need not be subgroups. More precisely, if G is an abelian group written additively then a (set) factorization of G is a representation ofGin the formG=H+K where H andK are subsets ofGand for each g ∈ G there is a unique pair h ∈ H and k ∈ K such that g = h+k.

Four years later, Haj´os [8] solved Minkowski’s problem by solving the equivalent group factorization problem. This initiated the investigation of set factorizations

Date: September 30, 2018.

2000Mathematics Subject Classification. 20K25, 20D40.

Key words and phrases. direct factorizations of groups, central products.

A.M. was supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K115799 and Grant No. ERC HU 15 118286, and by the J´anos Bolyai Research Schol- arship of the Hungarian Academy of Sciences. The work of A.M. on the project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 741420). He also received funding from ERC 648017.

1

of abelian groups. The interested reader is referred to the book [14] by Szab´o and Sands for a comprehensive account of problems, techniques, results and applications of this field.

In the present paper we consider set factorizations of a general group. For abelian groups our definition coincides with the one described above. Some specialized results which apply to our definition appeared in [9]. Another related concept which is studied in the literature, is the concept of a logarithmic signature. This concept found use in the search for new cryptographic algorithms which are based on finite non-abelian groups, and this application motivates the bulk of the available results.

A logarithmic signature of a group Gis a sequence [α1, ..., αs] of ordered subsets αi⊆G(not necessarily normal) such that each g∈Ghas a unique representation g =g1· · ·gs withgi ∈αi for all 1 ≤i ≤s. The first proposal for a cryptosystem which is based on logarithmic signatures can be found in [12]. Results on short logarithmic signatures can be found in [10].

Definition 1.1. LetGbe a group, and letX andY be two non-empty subsets ofG.

We shall say that the setwise product XY is direct, and denote this fact by writing X×Y for the setXY, if bothX andY are normal inG, and if everyg∈XY has a unique representationg=xy withx∈X andy∈Y.

We shall say that the groupGhas aset-direct factorization(decomposition) or is aset-direct product, ifG=X×Y for someX, Y ⊆G. A set-direct factorization of Gwill be called non-trivial, if and only if none ofX orY is a singleton consisting of a central element (whereby the second factor must beG). Furthermore, following [14, p.5], we shall say that a subsetX of a groupGisnormalized if 1G ∈X, and that the set-direct factorization G = X×Y is normalized if both X and Y are normalized. We shall show (Remark 2.10) thatZ(G)×Z(G) acts on the set of all direct factorizations of Gand that each orbit of this action contains at least one normalized factorization.

Our main result is a necessary and sufficient condition for a group G and an unordered pair X, Y of normal subsets ofGto satisfyG=X×Y. This condition involves a certain central subgroupZ ofG, and a family of set-direct factorizations ofZ. Recall (see Section 2.2) that a groupGis a central product of two subgroups M andN ifG=M N, andM andN centralize each other. In this caseM andN are normal inGandZ:=M ∩N is central inG. We writeG=M◦ZN.

Theorem 1.2. LetGbe a group and letX, Y be normal subsets ofG. SetM =⟨X⟩, N =⟨Y⟩, Z :=M ∩N. For every m∈M set Xm:=(

m⁻¹X)

∩Z and for every n∈N set Y_n :=(

n⁻¹Y)

∩Z. ThenG=X ×Y if and only if the following two conditions are met.

(a) G=M◦ZN.

(b) Z=Xm×Yn for every m∈M andn∈N.

Corollary 1.3. Simple groups have no non-trivial set-direct factorization.

Condition (b) of Theorem 1.2 specifies a family of set-direct factorizations ofZ, and the next definition characterizes the families of set-direct factorizations of Z that arise in this way. Recall that if H is an abelian group, andS ⊆H then the kernel of S in H is defined byKH(S) :={h∈H|hS=S} (we also writeK(S) if H is clear from the context). One can easily check that K_H(S) is a subgroup of H.

Definition 1.4. LetZ be an abelian group and letM={Mi}i∈I andN ={Nj}_j∈J be two multisets (i.e., repetitions allowed) of subgroups of Z. An MN-direct fac- torization system of Z is a pair of multisets A = {A_i}_i∈I and B = {B_j}_j∈J of subsets ofZ which satisfy for alli∈I andj∈J

Z=A_i×B_j,M_i≤K_Z(A_i),N_j ≤K_Z(B_j).

To make the connection between Condition (b) of Theorem 1.2 and Definition 1.4, we need the following fact (see Section 2.1, Lemma 2.1). Let G be a group, N EG, andZ≤N a central subgroup ofG. ThenZ acts by multiplication on the set ΩN of all conjugacy classes of Gcontained inN.

Theorem 1.5. LetGbe a group and letX, Y be normal subsets ofG. SetM =⟨X⟩, N = ⟨Y⟩, Z := M ∩N, and assume that G =M ◦Z N. For each m ∈ M and n ∈N let Mm and Nn be, respectively, the stabilizers of the conjugacy classes of m and n with respect to the multiplication action of Z. Set M = {Mm}m∈M

and N = {Nn}n∈N. Then G = X ×Y if and only if the pair of multisets ({Xm}_m∈M,{Yn}_n∈N)

is anMN-direct factorization system ofZ, where for every m∈M and for everyn∈N, we set Xm:=(

m⁻¹X)

∩Z andYn:=( n⁻¹Y)

∩Z.

The next theorem shows that starting from a central product G = M ◦Z N and an appropriate set-direct factorization system ofZ, one can obtain a set-direct decomposition ofG. We denote byO(Ω_M) andO(Ω_N) the set of allorbits of the multiplication action ofZ on Ω_M and Ω_N respectively. Note (Lemma 2.1 part 3) that the stabilizer of an orbit is the stabilizer of any conjugacy class belonging to the orbit.

Theorem 1.6. Let G = M ◦Z N. Set I := O(ΩM) and J := O(ΩN). For each i ∈I and j ∈ J let Mi and Nj be the stabilizers of the orbits i and j. Set M:={Mi}i∈I and N :={Nj}_j∈J. Assume that A:={Ai}i∈I andB:={Bj}_j∈J is anMN-direct factorization system ofZ. For eachi∈I andj∈J fix conjugacy classesCi andDj belonging to the orbitsiandj respectively. ThenG=X×Y is a set-direct decomposition of G, where

X := ∪

i∈I

AiCi andY := ∪

j∈J

BjDj.

Thus, the set-direct factorizations of a general group G can be obtained, in principle, from the knowledge of the central subgroups of G, the central product decompositions ofGin which they are involved (each centralZ ≤Gis involved in at least one such decomposition -G=G◦ZZ), the stabilizers of their multiplication action on the set of conjugacy classes of Gcontained in the factors of the central products, and the associated factorization systems.

Now we consider two special cases of this general observation. The first case is when one of the factors is a group (and then the second factor is a normal transversal for this group). We shall say, given a groupG, thatz∈Z(G) issemi- regular ifzC ̸=Cfor every conjugacy classCofG. In the followingk(G) denotes the number of conjugacy classes ofG.

Theorem 1.7. LetGbe a group and letX andY be normal subsets ofG. Suppose that X is a group. Set N := ⟨Y⟩ and Z := X ∩N, and for each n ∈ N set Y_n:=(

n⁻¹Y)

∩Z. Then the following two conditions are equivalent:

(1) G=X×Y.

(2) G=X◦ZN and|Yn|= 1 for alln∈N.

Furthermore, if either of (1) or (2) holds,Y is a normal transversal ofZ inN andZ acts semi-regularly onΩN. IfN is finite then k(N) =k(Z)k(N/Z).

Corollary 1.8. LetG:=M◦ZN and suppose that Z has a normal transversalY in N. Then G=M ×Y. In particular, if N is an abelian group, and Y is any transversal ofZ inN, thenG=M×Y.

Here we prove Corollary 1.8 by showing directly how the conditions of Definition 1.1 are fulfilled. In Section 3 we give a second proof of Corollary 1.8 which relies on its connection with Theorem 1.7.

Proof of Corollary 1.8. SinceG=M N,N =ZY andZ≤M we getG=M ZY = M Y. If m1y1 =m2y2 form1, m2 ∈ M and y1, y2 ∈ Y theny1y⁻₂¹ ∈M whereby y1y₂⁻¹ ∈ M ∩N = Z and hence y1 =y2 and m1 = m2. This proves the unique factorization property required in Definition 1.1, and henceG=M×Y.

In the second special case we consider, a single set-direct factorization of an abelian group Z induces a set-direct factorization of G = M ◦Z N. Our non- standard commutator notation is explained in the first paragraph of Section 2.

Theorem 1.9. LetG=M◦ZN be a group, andZ =X0×Y0a set-direct decompo- sition ofZ. Assume that [M, M]∩Z⊆KZ(X0)and[N, N]∩Z⊆KZ(Y0). Then Ghas a set-direct factorizationG=X×Y withX⊆M,Y ⊆N,X∩Z =X₀and Y ∩Z=Y₀.

Corollary 1.10. Let G be a group with a non-trivial central element z of prime power order which is not a prime. Suppose that z is semi-regular. Then G has a non-trivial set-direct factorization. Furthermore, ifGis also perfect, then Ghas a non-trivial normalized (see Remark 2.10) set-direct factorization such that none of the factors is a group.

Concrete examples of non-trivial set-direct factorizations of the type described by Corollary 1.10, whereGis a non-abelian finite quasi-simple group, are provided by the following theorem whose proof rests on the work of Blau in [3].

Theorem 1.11. Let G be a finite quasi-simple group. Then G has a non-trivial normalized set-direct decomposition if and only if G/Z(G) ∼= P SL(3,4) and 8 divides |Z(G)|. Moreover, Ghas a non-trivial normalized set-direct decomposition such that none of the two factors is a group if and only if G/Z(G)∼=P SL(3,4) and16divides |Z(G)|.

2. Notation and Background Results

Let G be any group. For x ∈ G we denote the conjugacy class of xin G by x^G. For any normal subset S ofG let ΩS denote the set of all conjugacy classes of G contained in S. We set k(G) := |ΩG| in the case that G is finite. For any two subsets A and B of G, we denote by [A, B] the set of all commutators [a, b] := a⁻¹b⁻¹ab where a and b vary over all elements of A and B respectively.

Note that for subgroupsA and B our notation differs from the common practice to denote by [A, B] the subgroup generated by all [a, b] witha∈A and b∈B. If A={a}we may write [a, B] for [{a}, B] and similarly [A, b] := [A,{b}].

2.1. The action of a central subgroup on conjugacy classes. In this sub- section we summarize basic properties of the multiplication action of a central subgroupZ≤Gon the conjugacy classes ofG.

Lemma 2.1. Let Gbe a group,N EG, andZ ≤N a central subgroup of G.

(1) Z acts by multiplication on Ω_N, namely, for any z ∈ Z and D ∈ Ω_N we havezD=Dz∈Ω_N. Denote this action byα.

(2) For any D∈Ω_N denote byZ_D ≤Z the stabilizer of D with respect toα.

Thenz∈Z_D if and only if there exists somed∈D such thatdz∈D.

(3) For any D ∈ Ω_N denote by O_D the orbit of D under α. Then for any D1, D2∈OD we have ZD₁=ZD₂.

(4) Let n∈N and letD=n^G. ThenZD= [n, G]∩Z.

(5) LetY be a normal subset ofGcontained inN. Letn∈N and letD=n^G. SetYn :=n⁻¹Y ∩Z. IfYn is non-empty then ZD ≤K(Yn) (equivalently, Yn is a union of cosets ofZD in Z).

(6) Suppose that G is a finite group. Then k(G/Z) is equal to the number of orbits of the multiplication action of Z on ΩG. It follows that Z acts semi-regularly on ΩG if and only if k(G) =k(Z)k(G/Z).

Proof. (1) Note that N is the union of all elements of ΩN and hence ΩN ̸=∅. LetD∈ΩN andz∈Z. Since Z≤N we haveDz⊆N. We have to show thatDzis also a conjugacy class ofG. Lety∈Dzandg∈G. Then there existsd∈D such thaty=dz. Using the fact that Z is central, we have

y^g= (dz)^g=d^gz∈Dz.

Thus, Dz is a normal subset of G. Now suppose that d1, d2 ∈ D. Then there existsg ∈Gsuch thatd2=d^g₁, and henced2z = (d1z)^g. Thus, any two elements inDzare conjugate inG. This completes the proof that Dz is also a conjugacy class ofG.

(2) One direction is trivial. In the other direction let d ∈ D be such that dz∈D. Then D∩Dz ̸=∅ and since both sets are conjugacy classes this forcesDz=D.

(3) SinceZ acts transitively onOD, the stabilizersZD₁ andZD₂ are conjugate inZ, and sinceZ is abelian, this impliesZD1=ZD2.

(4) Let z ∈ Z_D ≤ Z. Then nz ∈ D. This implies that there exists x ∈ G such thatnz =x⁻¹nx. Hence z=n⁻¹x⁻¹nx= [n, x]∈[n, G]. Therefore z∈[n, G]∩Z. Conversely, letz∈[n, G]∩Z. Then there existsx∈Gsuch that z= [n, x] =n⁻¹x⁻¹nx. It follows that nz =x⁻¹nx∈D. By part 2, this impliesz∈Z_D.

(5) We have to showY_nZ_D=Y_n. LetYe_n:=nY_n=Y∩nZ. SetU :=Y∩DZ.

Then, sincenZ ⊆DZ we haveYen =U∩nZ. On the other hand, sinceY is normal in G, we get that U is a union of classes in OD, and hence, by part 3,U ZD=U. This together with (nZ)ZD=nZ impliesYenZD=Yen. Finally, this impliesYnZD=n⁻¹YenZD=n⁻¹Yen=Yn.

(6) We first prove thatk(G/Z) is equal to the number of orbits of the multipli- cation action ofZ on ΩG. LetD∈ΩG. It is easy to check that the setDZ, viewed as a set of cosets ofZ inGis a conjugacy class ofG/Z. By part 1, DZ can also be viewed as a disjoint union of conjugacy classes ofGwhich form one orbit under the multiplication action ofZ. Hence ifDe∈Ω_G then

eitherDe ⊆DZorDZe is a distinct conjugacy class ofG/Z. Moreover, every conjugacy class of G/Z is of the form DZ for some D ∈ΩG. Therefore, k(G/Z) is equal to the number of orbits of the multiplication action ofZon Ω_G. Now, the length of each orbit is at most|Z|, andk(G) equals the sum of the lengths of all of the Z-orbits. Since the action of Z is semi-regular if and only if all of the lengths equal|Z|=k(Z), we get that the action is semi-regular if and only ifk(G) =k(Z)k(G/Z).

Remark 2.2. We make two notational remarks in relation to Lemma 2.1.

(1) Let G be a group and let X be a set of conjugacy classes of G. In some discussions (e.g., X is an orbit of Z) it is convenient to abuse notation and let X stand also for the normal G-subset ∪

C∈X

C. Conversely, if X is a normal subset ofGwe may use the same letterX to denote the set of all conjugacy classes of G which are contained in X. We trust the reader to figure out the correct interpretation from the context.

(2) In view of the third claim of Lemma 2.1, we also write ZO_C instead ofZC. 2.2. Central products and their conjugacy classes. The following theorem is at the basis of the construction of central products of groups.

Theorem 2.3 ([6], Theorem 2.5.3). Let M, N, Z be groups with Z ≤Z(M), and suppose that there is an isomorphismθofZintoZ(N). Then, if we identifyZ with its image θ(Z), there exists a groupG of the formG=M N, with Z =M ∩N ≤ Z(G)such that M centralizesN.

Any groupGwith M, N, Z as in the theorem is said to be the central product ofM andN (with respect toZ), and we will write G=M◦ZN.

Next we consider the structure of the conjugacy classes of a central product G=M◦ZN, and the multiplication action ofZ on them (see Section 2.1).

Lemma 2.4. Let G be a group and M and N normal subgroups of G, such that G=M◦ZN, whereZ :=M∩N. LetC be a conjugacy class ofG. Then:

(1) There exist a conjugacy class C_M of M and a conjugacy class C_N of N such that C=CMCN.

(2) Let {(

C_M⁽ⁱ⁾, C_N⁽ⁱ⁾ )}

i∈I

be the set of all the distinct pairs of conjugacy classes C_M⁽ⁱ⁾∈Ω_M andC_N⁽ⁱ⁾∈Ω_N such thatC=C_M⁽ⁱ⁾C_N⁽ⁱ⁾ for alli∈I. Then each one of O_M :=

{ C_M⁽ⁱ⁾

}

i∈I

and O_N :=

{ C_N⁽ⁱ⁾

}

i∈I

is a single orbit of the multiplication action ofZ onΩM andΩN respectively.

(3) OC =OMON andZOC =ZOMZON.

(4) The equality O_C = O_MO_N from part 3 defines a bijection O(Ω_G) → O(Ω_M)×O(Ω_N).

Proof. 1. Letg ∈C. Then, since G=M N, there exist m∈M and n∈N such thatg=mn. Hence:

C=g^G={(mn)^xy|x∈M, y∈N}={m^xn^y|x∈M, y∈N}=m^Mn^N, where the third equality relies on the fact that M and N centralize each other.

Now we can takeC_M :=m^M andC_N :=n^N.

2. Let CM and CN be as in the first part. Note that for any z ∈ Z, C = (z⁻¹CM

)(zCN). On the other hand, suppose thatC=A1B1whereA1∈ΩM and B₁ ∈ Ω_N. Then there existm₁ ∈ A₁ and n₁ ∈ B₁ such that g = mn =m₁n₁. Hence z := m⁻₁¹m = n₁n⁻¹ ∈ Z and m₁ = z⁻¹m while n₁ = zn. This implies A₁=z⁻¹C_M andB₁=zC_N. We have proved:

(∗) {(

C_M⁽ⁱ⁾, C_N⁽ⁱ⁾ )}

i∈I

={(

z⁻¹C_M, zC_N)

|z∈Z} , which implies the claim.

3. LetCM andCN be as in the first part. We haveOM ={z1CM|z1∈Z} and ON ={z2CN|z2∈Z}. This gives

OMON = {z1CMz2CN|z1, z2∈Z}={z1z2CMCN|z1, z2∈Z}

= {zC|z∈Z}=O_C. For the second claim, letz∈ZC. Then

CMCN =C=zC = (zCM)CN.

This implies that the pairs (zC_M, C_N),(C_M, C_N) ∈ Ω_M ×Ω_N yield the same conjugacy class C. By (∗) there exist z^′ ∈ Z such that z^′−1zCM = CM and z^′CN = CN. Hence z^′ ∈ ZC_N and z^′−1z ∈ ZC_M, and since z = z^′(

z^′−1z) we get z ∈ZC_MZC_N. This proves that ZC ≤ZC_MZC_N. On the other hand, for any z1∈ZC_M andz2∈ZC_N we get

z₁z₂C=z₁C_Mz₂C_N =C_MC_N =C.

This proves thatZCMZCN ≤ZC and altogether we getZC =ZCMZCN.

4. By part 3,O_C =O_MO_N withO_M ∈O(Ω_M) and O_N ∈O(Ω_N). Moreover, the proofs of parts 2 and 3 show thatO_C uniquely determines (O_M, O_N), and that each pair (O_M, O_N) uniquely determines an orbitO_CofG-conjugacy classes under the multiplication action ofZ.

The following type of subset plays an important role in the analysis of the action of a central subgroup on the conjugacy classes of central products.

Definition 2.5. Let Gbe a group, Z a central subgroup ofGandKEG. We set Z_[K] := [K, K]∩Z.

Lemma 2.6. Let G be a group and M and N normal subgroups of G, such that G=M◦ZN, whereZ :=M∩N.

(a)

[M, M]∩[N, N] =Z_[M]∩Z_[N].

(b) Let I andJ be indexing sets such that O(Ω_M) :={X_i}i∈I andO(Ω_N) = {Y_j}j∈J. Then:

Z_[M]= ∪

i∈I

Z_Xi ,Z_[N]= ∪

j∈J

Z_Yj andZ_[G] = ∪

i∈I

∪

j∈J

Z_XiYj =Z_[M]Z_[N]. Proof. (a) SinceG=M◦ZN, we have [M, M]∩[N, N]⊆Zand hence, by Definition 2.5,

[M, M]∩[N, N] = [M, M]∩[N, N]∩Z =Z[M]∩Z[N]. (b) We prove Z_[N] = ∪

j∈J

ZY_j. First note that sinceM centralizesN, we have, for any n∈N, [n, G] = [n, N]. Let j ∈ J. By Lemma 2.1 parts (3) and (4) and

Remark 2.2 part 2, if n ∈ N belongs to a conjugacy class in the orbit Yj then ZYj = [n, G]∩Z= [n, N]∩Z. Therefore

∪

j∈J

Z_Yj = ∪

n∈N

([n, N]∩Z) =Z∩ ∪

n∈N

[n, N] =Z_[N]. The proof thatZ_[M] = ∪

i∈I

ZX_i and Z_[G] = ∪

i∈I

∪

j∈J

ZX_iY_j is similar, and Z_[G] = Z_[M]Z_[N] follows from Lemma 2.4 part (3).

2.3. Basic properties of a set-direct product. The following lemma states several equivalent conditions for the directness of a setwise product. Although the group is not assumed to be abelian the proof is essentially the same as that of [14, Lemma 2.2], and is therefore omitted.

Lemma 2.7. LetGbe a group, and letX andY be two non-empty normal subsets of G. The following conditions are equivalent.

(a) XY =X×Y.

(b) XX⁻¹∩Y Y⁻¹={1G}.

(c) {Xy|y∈Y} or{xY|x∈X} is a partition ofXY.

Furthermore, ifX andY are finite sets thenXY is direct if and only if|XY|=

|X| · |Y|.

Remark 2.8. LetGbe a group, and letX andY be two non-empty normal subsets of G. ThenXY =X×Y implies |X∩Y| ≤ 1. To see this observe that if a, b∈ X∩Y, thenab=baare two factorizations in X×Y, and hence, by uniqueness of factorization,a=b.

Lemma 2.9. Let Gbe a group, and let G=X×Y be a set-direct decomposition of G. Then:

(a) If C is a conjugacy class of Gcontained in X, and |C|>1 thenY ∩C= Y ∩C⁻¹=∅.

(b) There exists a central elementz ofG such thatz∈X andz⁻¹∈Y. (c) If z is any central element ofGthenG= (zX)×Y =X×(zY).

Proof. (a) Since |C| >1 then C⁻¹C =CC⁻¹ must contain non-trivial elements, and hence, ifY containsC orC⁻¹ we get a contradiction with Lemma 2.7(b).

(b) By part (a), if C is a conjugacy class, C ⊆X andC⁻¹⊆Y, then C must consist of a single central element. On the other hand, since 1G∈G, we must have at least one classCsuch thatC⊆X andC⁻¹⊆Y.

(c) Since z is central, the normality of X implies the normality of zX and (zX)⁻¹(zX) =X⁻¹X. Similar claims hold whenX is replaced byY. Now apply Lemma 2.7.

Remark 2.10. In view of Lemma 2.9(c),Z(G)×Z(G)acts on the set of all direct factorizations ofGviaX×Y 7−→(z1X)×(z2Y)wherez1, z2∈Z(G)and by Lemma 2.9(b), each orbit of this action contains at least one normalized factorization. Note that ifX ⊆Gis normalized andX is not a subgroup ofGthengXis not a subgroup of Gfor any g∈G. For suppose by contradiction thatgX is a subgroup for some g ∈ G. Then, since 1_G ∈ X we get that g ∈ gX and hence g⁻¹ ∈gX, implying gX=g⁻¹(gX) =X, wherebyX is a subgroup - a contradiction.

Lemma 2.11. LetGbe a group, and letA, B, C be three non-empty normal subsets ofG. Suppose that the productsABand(AB)C are direct. Then the productsBC andA(BC) are also direct.

Proof. First we show that A(BC) is direct. For this we have to show that if a1, a2∈A,b1, b2∈B andc1, c2∈C and

a1(b1c1) =a2(b2c2) ,

thena1=a2, andb1c1=b2c2. By associativity, (a1b1)c1= (a2b2)c2. Since (AB)C is direct, we getc₁=c₂anda₁b₁=a₂b₂. SinceABis direct we further geta₁=a₂ and b₁ =b₂. It follows thatb₁c₁ = b₂c₂. Now we show that BC is direct. Let b₁, b₂ ∈ B, c₁, c₂ ∈ C and suppose that b₁c₁ = b₂c₂. Multiplying on the left by somea∈A(recall that by assumptionA̸=∅) and using associativity, we have

(ab1)c1= (ab2)c2.

Since (AB)C is direct, this givesc1=c2 andab1=ab2, which yieldsb1=b2. 3. Set-direct factorizations of groups

In this section we prove the various conditions stated in the Introduction for set-direct decompositions of groups. We begin by showing that if the product of two subsets of a group is direct then they must centralize each other.

Theorem 3.1. Let Gbe a group and let X andY be two normal subsets of G. If XY =X×Y then[X, Y] ={1G}.

Proof. LetX andY be two normal subsets ofG, and assume thatXY =X×Y. Letx∈X,y∈Y andt:=xy. It is clear thatCG(x)∩CG(y)≤CG(t). We prove that CG(t)≤CG(x)∩CG(y). Leth∈CG(t). Then xy =t =t^h =x^hy^h. Since x^h ∈ X and y^h ∈ Y, we have, by uniqueness of factorization, that x^h = x and y^h=y, implyingh∈CG(x)∩CG(y), andCG(t) =CG(x)∩CG(y). In particular, t ∈ CG(x)∩CG(y). Hence t commutes with bothx and y. But t = xy implies y=x⁻¹t. Using the fact thatxcommutes with bothx⁻¹and twe get thatxand y commute.

Using Theorem 3.1, we obtain Theorem 1.2 as a consequence of an apparently more general statement.

LetG be a group and let X and Y be subsets of G. We write G=X×cY if every elementg∈Gcan be uniquely expressed asg=xy wherex∈X andy∈Y, and ifX centralizesY. Clearly, for an abelianG,G=X×cY and G=X×Y is the same, and, in general,G=X×Y impliesG=X×cY.

Theorem 3.2. LetGbe a group and letX andY be subsets of G. SetM :=⟨X⟩, N :=⟨Y⟩andZ :=M∩N. For everym∈M andn∈N setX_m:=(

m⁻¹X)

∩Z andYn:=(

n⁻¹Y)

∩Z. ThenG=X×cY if and only if (a) G=M◦ZN; and

(b) Z=Xm×Yn for every m∈M andn∈N.

Proof. Assume thatG=X×cY. ThenM is centralized byY and henceM EG.

Similarly,N EG. Furthermore,M centralizesN. Thus (a) follows.

Now we prove (b). FromG=X×cY and by (a) we haveG=XY =M ◦ZN. Furthermore,G=X×cY andN =⟨Y⟩imply that ncentralizesX for alln∈N. SimilarlymcentralizesY for allm∈M. Hence, for allm∈M andn∈N we have

G=m⁻¹n⁻¹XY =m⁻¹Xn⁻¹Y, and this implies G = (

m⁻¹X)

×c

(n⁻¹Y)

for allm ∈M and n ∈ N (note that uniqueness of factorization follows from that of G =X ×cY). In particular, for all z ∈ Z there exist unique x ∈ X and y ∈ Y such that z = (

m⁻¹x) ( n⁻¹y)

. This gives m⁻¹x = y⁻¹nz. As the l.h.s. is in M and the r.h.s. is in N we get m⁻¹x∈(

m⁻¹X)

∩Z=Xmand similarly,n⁻¹y∈Yn. This provesZ =Xm×Yn. Conversely, assume that conditions (a) and (b) hold true. By (a) we get that X ⊆M centralizesY ⊆N. Letg∈G. By (a) there existm∈M andn∈N such thatg=mn. Using (b) and the fact thatXmandYn are central we get:

gZ= (mn)XmYn= (mXm) (nYn) = (X∩mZ) (Y ∩nZ)⊆XY.

Since g ∈ gZ this implies the existence of x ∈ X and y ∈ Y such that g =xy.

It remains to prove uniqueness of representation. Suppose that we also havex1∈ X and y1 ∈ Y such that g = x1y1. Then xy = x1y1 implying x⁻₁¹xy = y1. Since x⁻₁¹x ∈M and y ∈N, this implies yx⁻₁¹x=y₁ from which x⁻₁¹x=y⁻¹y₁ follows. Since the l.h.s. of the last equality is in M and the r.h.s. is in N we get x⁻₁¹x =y⁻¹y1 ∈ Z. It now follows that x⁻₁¹x∈ Xx₁ and y⁻¹y1 ∈ Yy, hence x⁻₁¹x=y⁻¹y1 ∈ Xx₁ ∩Yy. Now, since x1 ∈ X we have 1G ∈ Xx₁ and similarly 1G ∈Yy. Therefore 1G∈Xx₁∩Yy. By (b), the productXx₁Yy is direct, therefore, by Remark 2.8,Xx₁∩Yy ={1G}. This impliesx1=xandy1=y.

Proof of Theorem 1.2. Combine Theorems 3.1 and 3.2.

Proof of Theorem 1.5. Since condition (a) of Theorem 1.2 holds by assumption, we have that G=X×Y if and only if Z =Xm×Yn for every m∈M and n∈N. Hence, by Definition 1.4, it remains to show that for any m∈ M and n∈ N we haveM_m≤K(X_m) andN_n≤K(Y_n). This is immediate from Lemma 2.1 (5).

Remark 3.3. There is a considerable redundancy in the description of the fac- torization system (

{Xm}m∈M,{Yn}n∈N

) of Z used in Theorem 1.5. First of all, since Z is central we get that Xm₁ = Xm₂ for any two conjugate m1, m2 ∈ M, or equivalently, Xm depends only on the conjugacy class of min G. Furthermore, suppose that C1 andC2 are two conjugacy classes ofM which belong to the same Z-orbit. Let mi ∈Ci fori= 1,2 and let c12∈Z be such that C2=c12C1. Then by Lemma 2.1 (3) we have M_m1 =M_m2. Furthermore, we have X_m2 =c⁻₁₂¹X_m1. Since similar claims hold true for theY_nit follows that one can replace the indexing setsM andN by, respectively, the set of orbits of the multiplication action ofZ on Ω_M and on Ω_N, and replace(

{X_m}_m∈M,{Y_n}_n∈N)

by a ”trimmed” factorization system.

Theorem 3.4. LetZ be an abelian group and letM={Mi}_i∈I andN ={Nj}_j∈J be two multisets of subgroups ofZ. Let(A,B)be anMN-direct factorization system of Z whereA={Ai}i∈I andB={Bj}_j∈J. Then:

(a) For any i∈ I and j ∈ J, any two distinct elements a1, a2 ∈ Ai belong to distinct cosets ofN_j in Z and any two distinct elementsb₁, b₂∈B_j belong to distinct cosets ofM_i inZ.

(b) Suppose thatZ is finite. Then(A,B)has to satisfy the following arithmeti- cal conditions:

(3.1) |Z|=|Ai| |Bj|,∀1≤i≤r,∀1≤j≤s, which imply

(3.2) |A₁|=· · ·=|A_r|, |B₁|=· · ·=|B_s|, and also

(3.3) lcm (|M1|, ...,|Mr|) divides |A1| and lcm (|N1|, ...,|Ns|) divides |B1|. Proof. (a) We prove the first part of the claim. The proof of the second part

is essentially the same. Suppose, by contradiction, that there exist i ∈I andj∈J, and two elementsa1̸=a2∈Ai which belong to the same coset ofN_j in Z. Thena₁ =n₁xanda₂ =n₂x, wheren₁ ̸=n₂ are elements of N_j andx∈Z. SinceB_j is a non-empty union of cosets ofN_j, there exists somey∈Z, such thatN_jy⊆B_j. Henceb₁:=n₁y andb₂:=n₂ybelong to B_j anda₁b₂ =a₂b₁ are two distinct factorizations in A_i×B_j of the same element - a contradiction.

(b) Equation 3.1 is immediate from Lemma 2.7. For Equation 3.3 note that

|K(A)| divides |A| for any finite, non-emptyA. Hence, by Definition 1.4,

|Mi| divides |Ai| for all 1 ≤ i ≤ r. Similarly, |Nj| divides |Bj| for all 1≤j ≤s. Now Equation 3.3 follows from Equation 3.2.

Proof of Theorem 1.6. Fix arbitrary i ∈ I and j ∈ J. Since Mi ≤ K(Ai) we get that Ai is a union of cosets of Mi, and similarly, Bj is a union of cosets of Nj. Therefore we can write Ai = ∪

s

aisMi where the ais ∈ Z represent distinct cosets of Mi and similarly Bj = ∪

t

bjtNj, with bjt representing distinct cosets of Nj. Hence all the M-conjugacy classes aisCi, which are contained in AiCi, are distinct for distinct values ofs, and similarly, theN-conjugacy classesbjtDj which are contained inB_jD_j are distinct for distinct values oft. We claim that∪

s,t{a_isb_jt} is a transversal ofZij in Z, whereij is the uniqueZ-orbit formed by the product ofi andj (see Lemma 2.4 (3)). In the first place, usingZ_ij =M_iN_j (Lemma 2.4 (3)), we get

(∪

s,t

{aisbjt} )

Zij = (∪

s

aisMi

) (∪

t

bjtNj

)

=AiBj=Z.

Next, suppose that aisbjtZij = ais^′bjt^′Zij. This yields aisbjt ∈ (ais^′Mi) (bjt^′Nj) which implies the existence ofz1∈Miandz2∈Njsuch thataisbjt= (ais^′z1) (bjt^′z2).

Observe thatais^′z1∈Aiandbjt^′z2∈Bj. By uniqueness of factorization inAi×Bj

we getais=ais^′z1 andbjt=bjt^′z2. Ifs̸=s^′, we have, by our choice, thatais and a_is′ belong to distinctM_i cosets in contradiction toa_is=a_is′z₁. Hences=s^′ and similarly t =t^′. Now it follows that a_isb_jtC_iD_j are distinct conjugacy classes for distinct pairs (s, t), and∪

s,t{a_isb_jt}C_iD_j=ij. This shows that G=X×Y. The next theorem is needed for the proof of Theorem 1.7.

Theorem 3.5. LetN be a group andZa central subgroup ofN. Then the following conditions are equivalent: