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Register a new user$1$-product problems with congruence conditions in nonabelian groups
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Kevin Zhao
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Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let $mathsf{s}_{dmathbb{N}}(G)$ denote the smallest integer $ellin mathbb{N}_0cup {+infty}$ such that every sequence $S$ over $G$ of length $|S|geq ell$ 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: $mathsf{s}_{dmathbb{N}}(D_{2n})=2d+lfloor log_2nrfloor$, if $d$ is odd with $n|d$; $mathsf{s}_{dmathbb{N}}(D_{2n})=nd+1$, if $gcd(n,d)=1$. Furthermore, we also analysis the problem for metacyclic groups $C_pltimes_s C_q$ and obtain a result: $mathsf{s}_{kpmathbb{N}}(C_pltimes_s C_q)=lcm(kp,q)+p-2+gcd(kp,q)$, where $pgeq 3$ and $p|q-1$.
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