CHARACTER TABLE GROUPS AND EXTRACTED SIMPLE AND CYCLIC POLYGROUPS

Sara Sekhavatizadeh, Mohammad Mehdi Zahedi, Ali Iranmanesh

Abstract


Let ${G}$ be a finite group and $\hat{G}$ be the set of all irreducible complex characters of $G.$ In this paper, we consider $<\hat{G}, *>$ as a polygroup, where for each $\chi _{i} ,\chi_{j}\in \hat{G}$ the product $\chi _{i} * \chi_{j}$ is the set of those irreducible constituents which appear in the element wise product $\chi_{i} \chi_{j}.$ We call that $\hat{G}$ simple if it has no proper normal subpolygroup and
show that if $\hat{G}$ is a single power cyclic polygroup, then $\hat{G}$ is a simple polygroup and hence $\hat{S}_{n}$ and $\hat{A}_{n}$ are simple polygroups. Also, we prove that if $G$ is a non-abelian simple group, then $\hat{G}$ is a single power cyclic polygroup. Moreover, we classify $\hat{D}_{2n}$ for all $n.$ Also, we prove that $\hat{T}_{4n}$ and $\hat{U}_{6n}$ are cyclic polygroups with finite period.


Keywords


Character of group, hypergroup, polygroup, cyclic hyper- group, fundamental relation

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DOI: https://doi.org/10.22342/jims.26.1.742.22-36

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