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ABSTRACT
Let H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H′ be a further transfer Krull monoid over a finite abelian group G′ and suppose that ℒ(H) = ℒ(H′). We prove that, if
with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G′ are isomorphic.
1. Introduction and main result
Let H be an atomic unit-cancellative monoid. Then each non-unit a∈H can be written as a product of atoms, and if with atoms
of H, then k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a, and ℒ(H) = {L(a)∣a∈H} is called the system of sets of lengths of H (for convenience we set L(a) = {0} if a is an invertible element of H). Under a variety of noetherian conditions on H (e.g., H is the monoid of nonzero elements of a commutative noetherian domain) all sets of lengths are finite. Sets of lengths (together with invariants controlling their structure, such as elasticities and sets of distances) are a well-studied means for describing the arithmetic structure of monoids.
Let H be a transfer Krull monoid over a finite abelian group G. Then, by definition, there is a weak transfer homomorphism 𝜃:H→ℬ(G), where ℬ(G) denotes the monoid of zero-sum sequences over G, and hence ℒ(H) = ℒ(ℬ(G)). We use the abbreviation ℒ(G) = ℒ(ℬ(G)). By a result due to Carlitz in 1960, we know that H is half-factorial (i.e., |L| = 1 for all L∈ℒ(H)) if and only if |G|≤2. Suppose that |G|≥3. Then there is some a∈H with |L(a)|>1. If k,ℓ∈L(a) with k<ℓ and m∈ℕ, then L(am)⊃{km+ν(ℓ−k)∣ν∈[0,m]} which shows that sets of lengths can become arbitrarily large. Note that the system of sets of lengths of H depends only on the class group G. The associated inverse question asks whether or not sets of lengths are characteristic for the class group. In fact, we have the following conjecture (it was first stated in [Citation6] and for a detailed description of the background of this problem, see [Citation6], [Citation7, Section 7.3], [Citation8, page 42], and [Citation24]).
Conjecture 1.1.
Let G be a finite abelian group with D(G)≥4. If G′ is an abelian group with ℒ(G) = ℒ(G′), then G and G′ are isomorphic.
Note if D(G) = 3, then we have . The system of sets of lengths ℒ(G) is studied with methods from Additive Combinatorics. In particular, zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role. Most of these invariants are well-understood only in a very limited number of cases (e.g., for groups of rank two, the precise value of the Davenport constant D(G) is known and the associated inverse problem is solved; however, if n is not a prime power and r≥3, then the value of the Davenport constant
is unknown). Thus it is not surprising that most affirmative answers to the Characterization Problem so far have been restricted to those groups where we have a good understanding of the Davenport constant. These groups include elementary two-groups, cyclic groups, and groups of rank 2 (for recent progress we refer to [Citation1, Citation9]).
The first and so far only groups, for which the Characterization Problem was solved whereas the Davenport constant is unknown, are groups of the form , where r,n∈ℕ and 2r<n−2 (this is done by Geroldinger and Zhong [Citation12] and Zhong [Citation27]), which use a deep characterization of the structure of Δ*(G). In this paper, we go on to study groups of the form
and obtain the following theorem.
Theorem 1.2.
Let H be a transfer Krull monoid over a finite abelian group G with D(G)≥4. Suppose with r,n∈ℕ and ℒ(H) = ℒ(H′), where H′ is a further transfer Krull monoid over a finite abelian group G′. Then
If r≤n−3, then G≅G′.
If r≥n−1 and n is a prime power, then G≅G′.
This is made possible by introducing new invariants ρ(G,d) and ρ*(G,d) which are only depending on ℒ(G) (see Definitions 3.1 and 3.3). In Section 2 we gather the required background both on transfer Krull monoids as well as on Additive Combinatorics. In Section 3, we provide a detailed study of ρ(G,d) and ρ*(G,d). The proof of Theorem 1.2 will be provided in Section 4. The final section is concluding remarks and conjectures.
Throughout the paper, let be a finite abelian group with D(G)≥4, where
and
.
2. Background on transfer Krull monoids and sets of lengths
Our notation and terminology are consistent with [Citation7, Citation9]. For convenience, we set min∅ = 0. Let ℕ be the set of positive integers, let ℤ be the set of integers, let ℚ be the set of rational numbers, and let ℝ be the set of real numbers. For rational numbers a,b∈ℚ, we denote by [a,b] = {x∈ℤ∣a≤x≤b} the discrete, finite interval between a and b. If L⊂ℕ is a subset, then Δ(L) denotes the set of (successive) distances of L (that is, d∈Δ(L) if and only if d = b−a with a,b∈L distinct and [a,b]∩L = {a,b}) and ρ(L) = supL∕minL denotes its elasticity (for convenience, we set ρ({0}) = 1).
Let r∈ℕ and let be an r-tuple of elements of G. Then
is said to be independent if ei≠0 for all i∈[1,r] and if for all
an equation
implies that
for all i∈[1,r]. Furthermore,
is said to be a basis of G if it is independent and
. For every n∈ℕ, we denote by Cn an additive cyclic group of order n. Since
, r = r(G) is the rank of G and nr = exp(G) is the exponent of G.
2.1. Sets of lengths
By a monoid, we mean an associative semigroup with unit element, and if not stated otherwise we use multiplicative notation. Let H be a monoid with unit element 1 = 1H∈H. An element a∈H is said to be invertible (or an unit) if there exists an element a′∈H such that . The set of invertible elements of H will be denoted by H×, and we say that H is reduced if H× = {1}. The monoid H is said to be unit-cancellative if for any two elements a,u∈H, each of the equations au = a or ua = a implies that u∈H×. Clearly, every cancellative monoid is unit-cancellative.
Suppose that H is unit-cancellative. An element u∈H is said to be irreducible (or an atom) if u∉H× and for any two elements a,b∈H, u = ab implies that a∈H× or b∈H×. Let 𝒜(H) denote the set of atoms, and we say that H is atomic if every non-unit is a finite product of atoms. If H satisfies the ascending chain condition on principal left ideals and on principal right ideals, then H is atomic [Citation4, Theorem 2.6]. If a∈H∖H× and , where k∈ℕ and
, then k is a factorization length of a, and
2.2. Monoids of zero-sum sequences
Let G0⊂G be a non-empty subset. Then ⟨G0⟩ denotes the subgroup generated by G0. In Additive Combinatorics, a sequence (over G0) means a finite sequence of terms from G0 where repetition is allowed and the order of the elements is disregarded, and (as usual) we consider sequences as elements of the free abelian monoid with basis G0. Let
The sequence S is said to be
zero-sum free if 0∉Σ(S),
a zero-sum sequence if σ(S) = 0,
a minimal zero-sum sequence if it is a nontrivial zero-sum sequence and every proper subsequence is zero-sum free.
The set of zero-sum sequences is a submonoid, and the set of minimal zero-sum sequences is the set of atoms of ℬ(G0). For any arithmetical invariant ∗(H) defined for a monoid H, we write ∗(G0) instead of ∗(ℬ(G0)). In particular,
is the set of atoms of ℬ(G0),
is the system of sets of lengths of ℬ(G0), and so on. We denote by
2.3. Transfer Krull monoids
Let H be a atomic unit-cancellative monoid.
We say a monoid homomorphism 𝜃:H→B to an atomic unit-cancellative monoid B is a weak transfer homomorphism if it has the following two properties:
and
.
If a∈H, n∈ℕ,
and
, then there exist
and a permutation τ∈𝔖n such that
and
for each i∈[1,n].
Let 𝜃:H→B be a weak transfer homomorphism between atomic unit-cancellative monoids. It follows that for all a∈H, we have
and hence ℒ(H) = ℒ(B).
We say H is a transfer Krull monoid if one of the following equivalent conditions holds:
There exists a weak transfer homomorphism 𝜃:H→ℋ* for a commutative Krull monoid ℋ* (i.e., ℋ* is commutative, cancellative, completely integrally closed, and v-noetherian).
There exists a weak transfer homomorphism
for a subset G0 of an abelian group.
If the second condition holds, then we say H is a transfer Krull monoid over G0. If G0 is finite, then H is said to be a transfer Krull monoid of finite type.
We say a domain R is a transfer Krull domain (of finite type) if its monoid of cancelative elements R∙ is a transfer Krull monoid (of finite type).
In particular, commutative Krull monoids are transfer Krull monoids. Rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains are commutative Krull monoids with finite class group such that every class contains a prime divisor [Citation7, Section 2.11 and Examples 7.4.2]. Monoid domains and power series domains that are Krull are discussed in [Citation17, Citation2], and note that every class of a Krull monoid domain contains a prime divisor. Thus all these commutative Krull monoids are transfer Krull monoids over a finite abelian group.
However, a transfer Krull monoid need neither be commutative nor v-noetherian nor completely integrally closed. To give a noncommutative example, let 𝒪 be a holomorphy ring in a global field K, A a central simple algebra over K, and H a classical maximal 𝒪-order of A such that every stably free left R-ideal is free. Then H is a transfer Krull monoid over a ray class group of 𝒪 [Citation25, Theorem 1.1]. Let R be a bounded HNP (hereditary noetherian prime) ring. If every stably free left R-ideal is free, then its multiplicative monoid of cancelative elements is a transfer Krull monoid [Citation26, Theorem 4.4]. A class of commutative weakly Krull domains which are transfer Krull but not Krull will be given in [Citation10, Theorem 5.8]. Extended lists of commutative Krull monoids and of transfer Krull monoids, which are not commutative Krull, are given in [Citation6].
Let G0⊂G be a non-empty subset. For a sequence , we call
They were introduced by U. Krause in 1984 (see [Citation19]) and were studied under various aspects. For the relevance with the theory of non-unique factorizations, see [Citation21, Citation20, Citation22, Citation23] and [Citation7, Chapter 6].
Suppose , where r* is the total rank of G and
are prime powers, and set
It is easy to see that K*(G)≤K(G) and there is known no group for which inequality holds. For further progress on K(G), we refer to [Citation5, Citation15, Citation16, Citation18].
Lemma 2.1.
If G is a p-group, then K(G) = K*(G)<r(G).
Proof.
See [Citation7, Theorem 5.5.9].
A subset G0⊂G is called half-factorial if Δ(G0) = ∅. Otherwise, G0 is called non-half-factorial. Furthermore, the set G0 is called
minimal non-half-factorial if it is non-half-factorial and every proper subset
is half-factorial.
an LCN-set if k(A)≥1 for all A∈𝒜(G0).
We collect some easy or well known results which will be used throughout the manuscript without further mention.
Lemma 2.2.
Let G0⊂G be a non-empty subset. Then
G0 is half-factorial if and only if k(A) = 1 for all A∈𝒜(G0).
If G0 is an LCN-set, then
.
ℬ(G0) has accepted elasticity.
.
Δ(G) is an interval with minΔ(G) = 1.
If B∈ℬ(G), then ρ(L(Bk))≥ρ(L(B)) for every k∈ℕ.
If A∈𝒜(G), then {exp(G),exp(G)k(A)}⊂L(Aexp(G)).
Proof.
1. follows from [Citation7, Proposition 6.7.3] and 2. from [Citation7, Lemma 6.8.6].
2. see [Citation7, Proposition 6.7.3, Lemma 6.8.6].
3. follows from [Citation7, Theorem 3.1.4] and 4. from [Citation7, Section 6.3].
4. See [Citation7, Theorem 3.1.4, Section 6.3].
5. See [Citation11].
6. Let B∈ℬ(G) and k∈ℕ. Then maxL(Bk)≥kmaxL(B) and minL(Bk)≤kminL(B). It follows that .
7. Let A∈𝒜(G) and suppose , where ℓ∈ℕ and
. Then
Since A, are atoms, we obtain {exp(G),exp(G)k(A)}⊂L(Aexp(G)).
We need the following lemma.
Lemma 2.3.
Let be independent elements with the same order n, where r, n∈ℕ≥2. Then
If n≠r+1, then
If n = r+1, then
Proof.
1. See [Citation7, Proposition 6.8.2].
2. See [Citation7, Proposition 4.1.2.5].
3. Let and
. Let B∈ℬ(G0) and assume
Let
Then for every i∈I1, and for every j∈J1,
.
Note that for every and every
,
. Therefore
It follows that and hence n−2 | minΔ(G0). By 1., we obtain n−2 = r−1 = minΔ(G0).
If d∈ℕ and ℓ,M∈ℕ0, then a finite subset L⊂ℤ is called an almost arithmetical progression (AAP for short) with difference d, length ℓ, and bound M if
Lemma 2.4.
There exist constants such that for every A∈ℬ(G) with Δ(supp(A))≠∅, the set
is an AAP with difference minΔ(supp(A)), length at least 1, and bound M2.
Proof.
See [Citation7, Theorem 4.3.6].
Next, we recall the definition of the invariants Δ*(G) and Δ1(G) (see [Citation7, Definition 4.3.12]) in the Characterization Problem.
Let
We define
For every k∈ℕ, there exists some L∈ℒ(G) which is an AAP with difference d and length ℓ≥k.
Lemma 2.5.
Let k∈ℕ be maximal such that G has a subgroup isomorphic to . Then
.
.
.
.
Δ1(G) is an interval if and only if Δ*(G) is an interval if and only if r(G)+k≥exp(G)−1 or
.
Proof.
1. Follows from [Citation7, Corollary 4.3.16]
2. From [Citation14, Theorem 1.1].
3. See [Citation27, Proposition 3.7].
4. Follows from 1. and [Citation27, Theorem 1.1].
5. If Δ1(G) is an interval, then by 4 which implies that Δ*(G) is an interval by Zhong [Citation27, Theorem 1.1.2]. If Δ*(G) is an interval, then Δ1(G) is an interval by 1.. It follows by Zhong [Citation27, Theorem 1.1.2] that Δ*(G) is an interval if and only if r(G)+k≥exp(G)−1 or
.
Lemma 2.6.
Let G0⊂G with minΔ(G0)≥⌊exp(G)∕2⌋. Then
Let A∈𝒜(G0) with k(A)<1. Then
and ord(h) = exp(G), vh(A) = 1 for all h∈supp(A).
Let A∈𝒜(G0) with k(A)≥1. Then supp(A) is an LCN-set and
If G0 is a minimal non-half-factorial LCN-set with
, then |G0| = r(G)+1 and for every h∈G0, we have r(⟨G0∖{h}⟩) = r(G).
Let B∈ℬ(G) with ρ(L(B)) = ρ(G). Suppose
, where k = minL(B), ℓ = maxL(B), and
are atoms. Then k(Ui)≥1 for all i∈[1,k] and k(Vj)≤1 for all j∈[1,ℓ].
Proof.
1. Since {exp(G),exp(G)k(A)}⊂L(Aexp(G)), we have
Since minΔ(G0) | minΔ(supp(A)) and minΔ(G0)≥⌊exp(G)∕2⌋, it follows that
Therefore minΔ(G0) = minΔ(supp(A)) = exp(G)−exp(G)k(A) and hence .
Let g∈supp(A) such that . Then
, where B1∈ℬ(supp(A)). If there exists an atom A1 with k(A1)<1 such that A1 divides B1, then
. Therefore
and hence
. It follows by the minimality of
that ord(h) = exp(G) and vh(A) = 1 for all h∈supp(A).
2. Assume to the contrary that supp(A) is not an LCN-set. Then there exists A1∈𝒜(supp(A)) with k(A1)<1. It follows by 1 that for all h∈supp(A1) which implies that A1 | A, a contradiction. Thus supp(A) is an LCN-set and hence minΔ(supp(A))≤|supp(A)|−2.
3. By Geroldinger and Zhong [Citation14, Lemma 4.2], we have |G0| = r(G)+1 and r(⟨G0⟩) = r(G). If h∈⟨G0∖{h}⟩, then r(⟨G0∖{h}⟩) = r(G). Otherwise let d = min{k∈ℕ∣kh∈⟨G0∖h⟩} and hence (G0∖{h})∪{dh} is also a minimal non-half-factorial LCN-set with by Geroldinger and Halter-Koch [Citation7, Lemma 6.7.10]. It follows that r(⟨G0∖{h}⟩) = r(G).
4. Assume to the contrary that there exists i∈[1,k], say i = 1, such that k(U1)<1. Then exp(G)k(U1)<exp(G). Since , we infer
It follows by maxL(Bexp(G))≥exp(G)ℓ that , a contradiction.
Assume to the contrary that there exists j∈[1,ℓ], say j = 1, such that k(V1)>1. Then exp(G)k(U1)>exp(G). Since , we infer
It follows by minL(Bexp(G))≤exp(G)k that , a contradiction.
3. The invariants ρ(G,d), ρ*(G,d), and K(G,d)
First, we introduce new invariants which play a crucial role in the proof of Theorem 1.2 (see Proposition 3.5).
Definition 3.1.
Let d∈Δ1(G) and k∈ℕ. We define
Then is a decreasing sequence of positive real numbers and hence converges. We denote by ρ(G,d) the limit of
.
It follows by Geroldinger and Zhong [Citation13, Theorem 3.5] that ρ(G,1) = ρ(G) if and only if G is not a cyclic group of order 4,6 or 10.
Lemma 3.2.
Let G0⊂G be a subset with Δ(G0)≠∅. For every B∈ℬ(G0) with minΔ(G0)∈Δ(L(B)), we have ρ(G,minΔ(G0))≥ρ(L(B)).
Proof.
Let B∈ℬ(G0) with minΔ(G0)∈Δ(L(B)). By definition, L(B) is an AAP with difference minΔ(G0) and length at least 1. Therefore for every k∈ℕ, L(Bk) is an AAP with difference minΔ(G0) and length at least k. Thus for every k∈ℕ, by Lemma 2.2.6. Therefore ρ(G,minΔ(G0))≥ρ(L(B)).
Definition 3.3.
Let d∈Δ1(G). We define
Note that ρ*(G,1) = ρ(G) and K(G,1) = K(G). For every d∈Δ1(G), there always exists G0⊂G with G0 non-half-factorial such that d | minΔ(G0) by Lemma 2.5.1. Since ρ(G0)>1 and K(G0)≥1, we have ρ*(G,d)>1 and K(G,d)≥1. Furthermore, there exist with d | minΔ(G1) and d | minΔ(G2) such that
and K(G,d) = K(G2).
Lemma 3.4.
Let d∈Δ1(G). Then
ρ*(G,d)≥ρ(G,d).
for some k0∈ℕ with
.
.
Proof.
1. By the definition of Δ1(G) and d∈Δ1(G), for every k∈ℕ, we let Bk∈ℬ(G) be such that L(Bk) is an AAP with difference d and length at least k. Let
By Lemma 2.4, there exists a constant M such that minΔ(supp(B))∈Δ(L(BM)) for all B∈ℬ(G) with Δ(supp(B))≠∅. Since , we let k∈ℕ be large enough such that minL(Bk)≥M|𝒜(G)| and assume
Since minL(Bk)≥M|𝒜(G)|, we infer there exists i∈[1,s], say i = 1, such that t1≥M. Set I = {i∈[1,s]∣ti≥M} and . It follows by the choice of M that
Since L(Bk) is an AAP with difference d and is a subsequence of Bk, we obtain
Therefore d | minΔ(G0) and hence . Since
, there exists J⊂[1,maxL(Bk)] with |J|≥maxL(Bk)−M|𝒜(G)|D(G) such that
divides
. It follows by
that
Therefore
By definition, we infer which implies that ρ(G,d)≤ρ*(G,d).
2. Let G0⊂G with d | minΔ(G0) and . Then there exists B∈ℬ(G0) such that ρ(L(B)) = ρ*(G,d). Since supp(B)⊂G0, we infer minΔ(G0) divides minΔ(supp(B)) and hence d divides minΔ(supp(B)).
By Lemma 2.4, there exists a constant M such that minΔ(supp(B))∈Δ(L(BM)). It follows by Lemma 3.2 and Lemma 2.2.6 that
Thus the assertion follows by d divides minΔ(supp(B)).
3. For every k∈ℕ such that kd∈Δ1(G), we have by 1.. Therefore
. It follows by 2. that
.
Proposition 3.5.
Suppose ℒ(G) = ℒ(G′) for some finite abelian group G′ and let d∈Δ1(G). Then , ρ(G,d) = ρ(G′,d), and
.
Proof.
Since ℒ(G) = ℒ(G′), it follows by definition that and ρ(G,kd) = ρ(G′,kd) for every k∈ℕ such that kd∈Δ1(G). By Lemma 3.4.3, we obtain
.
Lemma 3.6.
Let d∈Δ1(G). Then ρ*(G,d)≥K(G,d). In particular, if d∈[1,r−1], then .
Proof.
Suppose G0⊂G with d dividing minΔ(G0) and K(G0) = K(G,d). Then there exists A∈𝒜(G0) such that k(A) = K(G,d)≥1. Since
In particular, if d∈[1,r−1], we let be independent elements of order n1. Set
and
. Then minΔ(G0) = d by Lemma 2.3.1.
Since is an atom with
, it follows that
Lemma 3.7.
Suppose . Let G0⊂G be a subset with d | minΔ(G0), where d∈Δ1(G) satisfies
. If A∈𝒜(G0) with k(A)>1, then
for some s∈ℕ.
In particular,
for some
, where
with
and
are pairwise distinct primes.
if nr is a prime power, then
.
if n1 is not a prime power, then K(G,r−1)>r.
Proof.
Let t = r−d and we start with the following claim.
Claim A.
Let B∈ℬ(G0) with for each g∈supp(B) and supp(B) is an LCN-set. Then there exists
with B0 is a product of atoms having cross number 1 and
for each g∈supp(B0) such that BB0 is a product of atoms having cross number 1.
Proof of Claim A.
Assume to the contrary that there exists a B∈ℬ(G0) with for each g∈supp(B) and supp(B) is an LCN-set, such that the assertion does not hold. Suppose |supp(B)| is minimal in all the counterexamples.
Set G1 = supp(B). If for all g∈G1, ord(g)∣vg(B), then B is a product of atoms having cross number 1, a contradiction. Therefore there exits such that
. Since
for each g∈supp(B), we infer
and
.
Let , where
and
are pairwise distinct primes. Let i∈[1,v] and
. Then
Therefore there exists such that
Since is an LCN-set, we have
Note that . We infer minΔ(supp(Bi)) = 0 and hence supp(Bi) is half-factorial. Therefore Bi is a product of atoms having cross number 1.
Since , there exist
such that
Therefore for some y∈ℕ and
. Note
for every i∈[1,v] and every
. Thus
for each g∈supp(C). Since |supp(C)|<|supp(B)|, it follows by the minimality of |supp(B)| that there exists
satisfying C0 is a product of atoms having cross number 1 and
for each g∈supp(C0), such that CC0 is a product of atoms having cross number 1. Let
. Then BB0 is a product of atoms having cross number 1, a contradiction to our assumption.
Let A∈𝒜(G0) be with k(A)>1. Then Lemma 2.6.2 implies that supp(A) is an LCN-set. Set and hence Claim A implies that there exist atoms
having cross number 1, where ℓ∈ℕ0, such that
is a product of atoms having cross number 1. Therefore
. Since d | minΔ(G0), we infer
. It follows that
for some s∈ℕ.
Now we begin to prove the “in-particular’’ parts.
1. For every j∈[1,u] and every m∈ℤ, we denote by ∥m∥j the least positive residue of m modulo , that is,
and
.
By definition of K(G,r−1), we have for some s∈ℕ. Let H be a subgroup of G with
We suppose that with
, where
and
,
. Then
Since , we infer
Therefore
Since , we infer r−1 | minΔ(G2) which implies that K(G,r−1)≥K(G2). Let
be the atom with
. Then
.
Since for some s∈ℕ, it follows that
.
2. If nr is a prime power, then K(G)<r by Lemma 2.1. It follows by 1. that for some s∈ℕ. Therefore
.
3. If n1 is not a prime power, then u≥2 and . It follows by 1. that K(G,r−1)>1+r−1 = r. □
Proposition 3.8.
Let s∈ℕ be maximal such that G has a subgroup isomorphic to . Suppose d∈Δ1(G) with
.
If d≥r, then
.
If d≥nr−1, then ρ*(G,d) = K(G,d).
Suppose r = nr−1≥3. If
for some prime p and k∈ℕ, then
. Otherwise, K(G,r−1) = ρ*(G,r−1).
Proof.
1. Since and d≥r, it follows by Lemma 2.5(items 3 and 4) that d>m(G) and
.
Let be independent elements with order nr and let
. Then
by Lemma 2.3.2. Since
is an atom, we infer
. Therefore
Let G0⊂G with d | minΔ(G0) be such that . Then there exists B∈ℬ(G0) with ρ(L(B)) = ρ*(G,d). Set
If there exists i∈[1,k] such that k(Ui)>1, then Lemma 2.6.2 implies that supp(Ui) is an LCN-set and hence minΔ(supp(Ui))≤m(G). Since
If there exists i∈[1,ℓ] such that k(Vi)<1, then Lemma 2.6.1 implies that . Therefore
for all i∈[1,ℓ]. It follows that
Then and hence
.
2. Let G0⊂G be such that d | minΔ(G0) and . If there exists an atom A∈𝒜(G0) such that k(A)<1, then Lemma 2.6.1 implies that
, a contradiction to |A|≥2. Thus G0 is an LCN-set. Let B∈ℬ(G0) such that
. Then
Let G0⊂G be such that d | minΔ(G0) and K(G0) = K(G,d). Then there exists an atom A∈𝒜(G0) such that k(A) = K(G,d)≥1. Since , we infer
3. Let r = nr−1≥3 and we proceed to prove the following claim.
Claim B.
Suppose ρ*(G,r−1)>K(G,r−1) and let G0⊂G be such that (r−1) | minΔ(G0) and ρ(G0)>K(G,r−1).
There exists g∈G0 with ord(g) = nr such that −g∈G0.
Let
with |G2| = r. If there exists a∈[1,nr−1] such that ag∈⟨G2⟩, then
,
, and G2 is a basis of G.
,
, where
and
is a basis of G, and
. In particular,
.
Proof of Claim B.
By Lemma 2.5.2, we infer that .
a. If G0 is an LCN-set, then for every B∈ℬ(G0), we have
b. Let E⊂G2 be minimal such that there exists a∈[1,nr−1] such that ag∈⟨E⟩ and let be minimal such that dgg∈⟨E⟩. Then there exists an atom V∈𝒜(E∪{g}) with
and |supp(V)|≤r+1. Let
, where T∈ℱ(E). Then
Note that for each , we have r−1 | dg+ℓ−2. Therefore
or (dg = 1 and
). We distinguish two cases.
Suppose . Let
such that (−g)Ti,
, are atoms, where
. Thus −g∈⟨E⟩ which implies that dg = 1 by the minimality of dg. The minimality of E implies that supp(Ti) = E for each i∈[1,nr−1]. Then for every h∈E,
. Therefore ord(h) = nr and
. It follows that
If k((−g)T1)<1, then Lemma 2.6.1 implies that T1 = g, a contradiction. Therefore
Since |E|≤|G2|≤r and r = nr−1, we have |E| = |G2| = r. Let . Then
implies that
is a basis of G,
, and
.
Suppose dg = 1 and . Note that V = gT and hence |T|≥2. We infer
. It follows by Lemma 2.6.2 that {−g}∪E is an LCN-set and minΔ({−g}∪E)≤|E|−1. Since (r−1) | minΔ({−g}∪E) and |E|≤|G2| = r, we have
Let be a minimal non-half-factorial LCN-set. Then
It follows by Geroldinger and Zhong [Citation14, Lemma 4.2] that |E1| = r+1 and hence . Therefore
Let and assume
If k(V)>1, then Lemma 2.6.2 and [Citation14, Lemma 4.5] imply
By the minimality of E = G2, we have for every m∈[1,nr−1] and every h∈G2, mg∉⟨G2∖{h}⟩. Note that nr≥4. Let . Then [Citation14, Lemma 4.4.1] implies that
Thus k(V) = 1 which implies that and
for each i∈[1,r]. It follows by k(V1)>1 and [Citation14, Lemma 4.3.2] that
is also an atom. Therefore
are independent and
. Since r(G) = r and exp(G) = nr, we infer
is a basis of G and
.
c. If for all W∈𝒜(G0), k(W)≤1, then for every B∈ℬ(G0), we have
Let W∈𝒜(G0) with k(W)>1. Then supp(W) is an LCN-set with minΔ(supp(W)) = r−1 by Lemma 2.6.2. Let G1⊂supp(W) be a minimal non-half-factorial subset. Then minΔ(G1) = r−1 and hence |G1| = r+1 by Geroldinger and Zhong [Citation14, Lemma 4.2.1]. Since {g,−g}⊄supp(W), we choose h∈G1 such that {g,−g}∩(G1∖{h}) = ∅. Lemma 2.6.3 implies r(⟨G1∖{h}⟩) = r. Thus there exists a∈[1,nr−1] such that ag∈⟨G1∖{h}⟩. Since |G1∖{h}| = r, it follows by a and b that and there exists a basis
of G such that
and
.
Assume to the contrary that there exists . After renumbering if necessary, we may assume that
, where t∈[1,r],
for each i∈[1,t] and
. Thus
. It follows by b that
and hence
, a contradiction. Therefore
.
We only need to prove which immediately implies that
.
Since , we obtain
Let B∈ℬ(G0) such that ρ(L(B)) = ρ(G0) and assume that
Note that . By Lemma 2.6.4, we have k(Ui)≥1 for all i∈[1,k] and k(Vj)≤1 for all j∈[1,ℓ]. Since
and
, substituting B by
, if necessary, we can assume that
,
for each i∈[1,k] and each j∈[1,ℓ].
Since there must exist j0∈[1,ℓ] such that , there must exists i0∈[1,k] such that
which implies that
for all j∈[1,ℓ]. If there exists i1∈[1,k] such that
, then
which implies that
Therefore
It follows by Lemma 3.6 that , a contradiction to ρ(G0)>K(G,r−1). If there exists j1∈[1,ℓ] such that
, then
which implies that
To sum up, we obtain
Let ,
, and J = {j∈[1,ℓ]∣Vj = g(−g)}. Then
Therefore and
. It follows that
.
We distinguish two cases to finish the proof.
Suppose that for some prime p and k∈ℕ with pk≥4. Then
by Lemma 3.7.2. Let
be a basis of G and
, where
. By Lemma 2.3.3, we have
. Since
It follows by Claim B that
Suppose that for any prime p and any k∈ℕ. Assume to the contrary that K(G,r−1)<ρ*(G,r−1). Then Claim B implies that
and ρ(G,r−1)<r. Since nr is not prime power, it follows by Lemma 3.7.3 that K(G,r−1)>r, a contradiction. □
4. Proof of main theorems
From now on, we assume G′ is a further finite abelian group with , where
and
. For convenience, we collect some necessary results which will be used all through the following two sections without further mention.
Lemma 4.1.
Suppose ℒ(G) = ℒ(G′) and d∈Δ1(G). Then
If G is isomorphic to a subgroup of G′, then G≅G′.
.
and
.
Proof.
1. By Geroldinger and Halter-Koch [Citation7, Proposition 7.3.1.3], we have D(G) = D(G′). It follows from [Citation7, Proposition 5.1.3.2 and 5.1.11.1] that G≅G′.
2. follows from [Citation7, Corollary 4.3.16] and [Citation14, Theorem 1.1.3].
3. See Proposition 3.5.
Theorem 4.2.
Suppose ℒ(G) = ℒ(G′). Then
If r≥nr−1 and n1≠2, then
.
If r≥nr−1, n1≠2, and nr is a prime power , then r(G) = r(G′) and
.
If r≤nr−3, then exp(G) = exp(G′) and
.
If
, then exp(G) = exp(G′) and r(G) = r(G′).
If r≤nr−3 and Δ*(G) is an interval, then exp(G) = exp(G′) and r(G) = r(G′).
Proof.
1. Assume to the contrary that r(G′)≠r(G) = r. Since , it follows that
. Let d = r−1. Then
. Since
by Lemma 3.6 and
by Proposition 3.8.1, it follows that
, a contradiction to n1≠2. Thus
and hence
.
2. Note that r≥nr−1≥2. If r = 2, then . Therefore D(G) = D(G′) = 5 and
. It follows that G≅G′. Thus we can assume r≥3.
By 1., we have . Since nr is a prime power and nr≥3, it follows by Lemma 3.7.2 that
. Then r≥3 and Propositions 3.8.2 and 3.8.2 imply that ρ*(G,r−1)<r and hence
. Therefore
If K(G,r−1) = ρ*(G,r−1) and , then
which infers
.
If K(G,r−1)<ρ*(G,r−1) and , then
, r = nr−1, and
by Proposition 3.8.3. Therefore
which infers
, a contradiction.
If K(G,r−1) = ρ*(G,r−1) and , then
, r = exp(G′)−1, and
by Proposition 3.8.3. Therefore
which infers
, a contradiction.
If K(G,r−1)<ρ*(G,r−1) and , then
by Proposition 3.8.3.
3. Note that nr≥4. Assume to the contrary that . Let
. Therefore
and Lemma 3.6 implies that
. Since d≥max{r,⌊nr∕2⌋}, Proposition 3.8.1 implies that
, a contradiction to
.
Thus . Since
, it follows that
.
4. By 3, exp(G) = exp(G′) and . Assume to the contrary that r(G)≠r(G′).
Suppose that r(G)>r(G′). Choose d = r−1. Therefore and Lemma 3.6 implies that
. Since
, Proposition 3.8.1 implies that
, a contradiction to
.
Suppose that r(G)<r(G′). Choose d = r∈[1,r(G′)−1]. Then and Lemma 3.6 implies that
. Since d≥max{r,⌊nr∕2⌋}, Proposition 3.8.1 implies that
, a contradiction to
.
5. By 3, we have and by 4, we can assume that
and
. Since Δ*(G) is an interval, we obtain that
is an interval by Lemma 2.5.5. Let k be maximal such that there exists a subgroup H of G with
and let k′ be maximal such that there exists a subgroup H′ of G′ with
. Then 2r≥r+k≥nr−2 and
by Lemma 2.5.5.
Assume to the contrary that r≠r(G′) and by symmetry, we can assume r<r(G′). Thus which implies that nr is even, nr = 2r+2, and r(G′) = r+1. Since Δ*(G) and
are intervals, it follows by Lemma 2.5.5 that
and k′≥r. Then G is a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1, a contradiction.
Proof of Theorem 1.2.
By definition of transfer Krull monoids, it follows that .
1. Let r≤n−3. If Δ*(G) is not an interval, then [Citation27, Theorems 1.1 and 1.2] implies that G≅G′.
Suppose Δ*(G) is an interval. Then Theorem 4.2.5 implies that n = exp(G′) and r = r(G′). Therefore G′ is isomorphic to a subgroup of G which implies that G≅G′ by Lemma 4.1.1.
2. If n = 2, then G is an elementary two-group and the assertion follows by Geroldinger and Halter-Koch [Citation7, Theorem 7.3.3]. We assume n≥3 is a prime power. Then Theorem 4.2.2 implies that r = r(G′) and . Therefore G is isomorphic to a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1.
5. Concluding remarks and conjectures
Throughout this section, let with D(G)≥4 and
, where
and
are pair-wise distinct primes.
Conjecture 5.1.
Suppose r≥n−1. Then the following hold:
for some s∈ℕ with gcd(s,n) = 1.
.
It is easy to see that C2 implies C1. By Lemma 3.7.1, we have for some s∈ℕ and if n is a prime power, then C2 holds.
Proposition 5.2.
Let G′ be a finite abelian group with ℒ(G′) = ℒ(G). If r≥n−1 and C1 holds, then G≅G′.
Proof.
If n is a prime power, then the assertion follows by Theorem 1.2.
Suppose n is not a prime power. Then r≥n−1≥5. Let ℒ(G) = ℒ(G′) and let with
and
. Then Theorem 4.2.1 implies that
. By Lemma 3.7.1 and Proposition 3.8(items 2. and 3.), we have
for some s∈ℕ. If
, then Lemma 3.7.1 implies that
for some s′∈ℕ. Thus
implies that
and hence
by gcd(s,n) = 1. Therefore G is isomorphic to a subgroup of G′ which implies that G≅G′ by Lemma 4.1.1.
Suppose . Then Lemma 3.6 implies that
and hence Proposition 3.8(items 2. and 3.) implies that
and
. Thus
implies that
. Since gcd(s,n) = 1, we have n = r+2, a contradiction.
Recall that for all finite abelian group G′ and there is known no group G′ with
.
Proposition 5.3.
Let G′ be a finite abelian group with ℒ(G′) = ℒ(G). If r≥max{(u−1)n+1,n}≥3 and K(G) = K*(G), then G≅G′.
Proof.
If u = 1, then the assertion follows by Theorem 1.2.2. Suppose u≥2. Note , where
. Since r≥(ω(n)−1)n+1, we have
. It follows by Lemma 3.7.1 that
, where s′∈ℕ. Therefore
. Since
, we have gcd(s,n) = 1. The assertion follows by Proposition 5.2.
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