Abstract
The complex fuzzy environment is an innovative tool to handle ambiguous situations in different mathematical problems. In this article, we commence the abstraction of (rho,eta)-complex fuzzy sets, (rho,eta)-complex fuzzy subgroupoid, (rho,eta)-complex fuzzy subgroups and describe important examples of the symmetric group under (rho,eta)-complex fuzzy sets. Additionally, we discuss the conjugacy class of the group with respect to (rho,eta)-complex fuzzy normal subgroups. We define (rho,eta)-complex fuzzy cosets and elaborate upon the certain operation of this analog to group theoretic operation. We prove that factors regarding the (rho,eta)-complex fuzzy normal subgroup form a group and establish an ordinary homomorphism. Moreover, we create the (rho,eta)-complex fuzzy subgroup of the factor group.