1-product problems with congruence conditions in nonabelian groups

2020/03/31 by Zhao, Kevin
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.14007

Abstract

Let G be a finite group and D_2n be the dihedral group of 2n elements. For a positive integer d, let s_dℕ(G) denote the smallest integer ℓ∈ ℕ₀∪ \+∞\ such that every sequence S over G of length |S|≥ ℓ has a nonempty 1-product subsequence T with |T|≡ 0 (mod d). In this paper, we mainly study the problem for dihedral groups D_2n and determine their exact values: s_dℕ(D_2n)=2d+\lfloor log₂n\rfloor, if d is odd with n|d; s_dℕ(D_2n)=nd+1, if gcd(n,d)=1. Furthermore, we also analysis the problem for metacyclic groups C_p\ltimes_s C_q and obtain a result: s_kpℕ(C_p\ltimes_s C_q)=lcm(kp,q)+p-2+gcd(kp,q), where p≥ 3 and p|q-1.

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