The large Davenport constant I: Groups with a cyclic, index 2 subgroup
The large Davenport constant I: Groups with a cyclic, index 2 subgroup
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DOI：
10.1016/j.jpaa.2012.09.004
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发表时间：
2012-11
影响因子：
0.8
通讯作者：
A. Geroldinger;D. Grynkiewicz
中科院分区：
文献类型：
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作者：
A. Geroldinger;D. Grynkiewicz
设G是一个有限群，其上写有乘法.对于G上的一个序列，我们指的是来自G的一个无序的有限项序列，允许项的重复，如果它的项可以被排序，使得它们的乘积是G的单位元，我们说它是一个乘积1序列。小达文波特常数d（G）是最大整数，使得G上有一个长度为的序列没有非平凡的乘积1子序列。大达文波特常数D（G）是最小积1序列的最大长度-这是一个不能分解为两个非平凡的积1序列的积1序列。很容易观察到d（G）+1 D（G），并且如果G是阿贝尔的，则等式成立。然而，对于非阿贝尔群，这些常数可以显著不同。设G有一个循环的指数为2的子群。然后Olson和白色的一个旧结果（可以追溯到1977年）暗示d（G）=12| G|如果G是非循环的，且d（G）=| G|-1，如果G是循环的。本文确定了这类群的大Davenport常数，证明了D（G）=d（G）+|G′|其中G′=[G，G]≤G是G的换位子群.
Let G be a finite group written multiplicatively. By a sequence over G, we mean a finite sequence of terms from G which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. The small Davenport constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constantD(G) is the maximal length of a minimal product-one sequence—this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that d(G)+1≤D(G), and if G is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Suppose G has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that d(G)=12|G| if G is non-cyclic, and d(G)=|G|−1 if G is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that D(G)=d(G)+|G′|, where G′=[G,G]≤G is the commutator subgroup of G.