Units in the group algebra $FS_{3}$ - Gupta, Abhinay Kumar; Sharma, R. K. (Ukrains’kyi Matematychnyi Zhurnal, 2026)
UDC 512.552We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p ≥ 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p mid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations.