Sumsets in dicyclic groups Q_4n and U_m,n

Abstract

Let G be an arbitrary group. For any subsets A and B of G, let $A*B=\{a*b;a\in A,b\in B\}$ where $\mathrm{`}*$’ is the binary operation on G. By ${\mu }_G(r,s)$, we denote the minimum cardinality of the set $A*B$, where $\left|A\right|=r$ and $\left|B\right|=s$. In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers $r, s$ with $1\le r,s\le g$, $k_G(r,s)\le {\mu }_G(r,s)$, where $k_G(r,s)={\min }_{d\in H(G)}\left\{\left(\lceil \frac{r}{d}\rceil +\lceil \frac{s}{d}\rceil -1\right)d\right\}$ and $H(G)$ is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group $D_q$ of index $q=p^v$. In this paper, we determine the upper bound for the size of ${\mu }_{Q_{4n}}(r,s)$ and ${\mu }_{U_{m,n}}(r,s)$, where $Q_{4n}=〈a,b:a^{2n}=e,b^2=a^n,ab=b{a}^{-1}〉$ is a dicyclic group of order 4n and $U_{m,n}=〈a,b:a^{2n}=e,b^m=e,ab{a}^{-1}=b^{-1}〉$ and verify the conjecture proposed by Eliahou et al. [7] for $Q_{4n}$ for n to be a prime power.

KEYWORDS:

• Dicyclic group
• dilates
• initial segment
• Minkowski
• sumsets

Additional information

Funding

Thankful to Dr MSG for their guidance and the research supported by SERB (MTR/2022/000331).