$1$-product problems with congruence conditions in nonabelian groups

Abstract

Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let ${\mathsf{s}}_{d\mathbb{N}}(G)$ denote the smallest integer $\ell \in {\mathbb{N}}_0\cup \{+\infty \}$ such that every sequence $S$ over $G$ of length $|S|\ge \ell$ has a nonempty $1$-product subsequence $T$ with $|T|\equiv 0$ (mod $d$). In this paper, we mainly study the problem for dihedral groups $D_{2n}$ and determine their exact values: ${\mathsf{s}}_{d\mathbb{N}}(D_{2n})=2d+\lfloor lo{g}_{2}n\rfloor$, if $d$ is odd with $n|d$; ${\mathsf{s}}_{d\mathbb{N}}(D_{2n})=nd+1$, if $gcd(n,d)=1$. Furthermore, we also analysis the problem for metacyclic groups $C_p{⋉}_s{C}_q$ and obtain a result: ${\mathsf{s}}_{kp\mathbb{N}}(C_p{⋉}_s{C}_q)=lcm(kp,q)+p-2+gcd(kp,q)$, where $p\ge 3$ and $p|q-1$.