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A quantitative aspect of non-unique factorizations: the Narkiewicz constants II

Weidong GaoYuanlin LiJiangtao Peng — 2011

Colloquium Mathematicae

Let K be an algebraic number field with non-trivial class group G and K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) denote the number of non-zero principal ideals a K with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F k ( x ) behaves, for x → ∞, asymptotically like x ( l o g x ) 1 / | G | - 1 ( l o g l o g x ) k ( G ) . In this article, it is proved that for every prime p, ( C p C p ) = 2 p , and it is also proved that ( C m p C m p ) = 2 m p if ( C m C m ) = 2 m and m is large enough. In particular, it is shown that for...

Unit groups of group algebras of some small groups

Gaohua TangYangjiang WeiYuanlin Li — 2014

Czechoslovak Mathematical Journal

Let F G be a group algebra of a group G over a field F and 𝒰 ( F G ) the unit group of F G . It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of F A 4 over any finite field of characteristic 3 and the structure of...

On the index of length four minimal zero-sum sequences

Caixia ShenLi-meng XiaYuanlin Li — 2014

Colloquium Mathematicae

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = ( n g ) · . . . · ( n l g ) where g ∈ G and n , . . . , n l i [ 1 , o r d ( g ) ] , and the index ind(S) is defined to be the minimum of ( n + + n l ) / o r d ( g ) over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong GaoYuanlin LiPingping ZhaoJujuan Zhuang — 2016

Colloquium Mathematicae

Let G be an additive finite abelian group. For every positive integer ℓ, let d i s c ( G ) be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine d i s c ( G ) for certain finite groups, including cyclic groups, the groups G = C C 2 m and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...

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