2018
DOI: 10.1007/s10998-018-00276-9
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On the algebraic and arithmetic structure of the monoid of product-one sequences II

Abstract: Let G be a finite group and G ′ its commutator subgroup. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a productone sequence if its terms can be ordered such that their product equals the identity element of G. The monoid B(G) of all product-one sequences over G is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if G is abelian (equiv… Show more

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Cited by 11 publications
(7 citation statements)
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“…where the intersection • (1) is taken over all finite abelian groups G with |G| ≥ 3, • (2) is taken over all non-half-factorial transfer Krull monoids H over finite abelian groups, and • (3) is taken over all finite groups with |G| ≥ 3. We recall that Equation (a) easily follows from Equation (1.1), Equation (b) is proved in [19,Section 3], and Equation (c) can be found in [27,Proposition 4.1]. Now we consider numerical monoids, where by a numerical monoid, we mean an additive submonoid of (N 0 , +) whose complement in N 0 is finite.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…where the intersection • (1) is taken over all finite abelian groups G with |G| ≥ 3, • (2) is taken over all non-half-factorial transfer Krull monoids H over finite abelian groups, and • (3) is taken over all finite groups with |G| ≥ 3. We recall that Equation (a) easily follows from Equation (1.1), Equation (b) is proved in [19,Section 3], and Equation (c) can be found in [27,Proposition 4.1]. Now we consider numerical monoids, where by a numerical monoid, we mean an additive submonoid of (N 0 , +) whose complement in N 0 is finite.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…We define a sequence over G to be an element of the free abelian monoid F (G), · , see Chapter 5 of [23], Section 3.1 of [10] or [16] for detailed explanation. Our notation of sequences follows the notation in the papers [21,25,30]. Note that, since G is a multiplicative group, we denote by ∅ the unit element of F (G) which is called the empty sequence.…”
Section: Preliminariesmentioning
confidence: 99%
“…• minimal product-one sequence if 1 G ∈ π(S) and S cannot be factored as a product of two non-trivial product-one subsequences. For recent study on the algebraic and arithmetic structure of product-one sequence for non-abelian groups, we refer to [30]. Using the above concepts, let…”
Section: Preliminariesmentioning
confidence: 99%
“…Both statements generalize to rings with zero-divisors ([12, Theorem 3.5 and Section 4]). We refer to [17,1,6,20] for C-monoids, that do not stem from ring theory, and to [18] for a more general concept.…”
Section: Background On C-monoidsmentioning
confidence: 99%
“…But this difference in complexity stems from the fact that the structure of the class semigroup of a C-monoid H can be much more intricate than the structure of the class group C( H) of its complete integral closure. We provide an explicit example of a half-factorial seminormal C-monoid (Example 4.3) and we also refer to the explicit examples of class semigroups given in [20,Section 4].…”
Section: A Characterization Of Half-factorialitymentioning
confidence: 99%