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Published online by Cambridge University Press: 20 November 2018
Let $p$, $q$ be prime numbers with ${{p}^{2}}\,<\,q,\,n\,\in \,\mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $p{{q}^{n}}$ over an algebraically closed field of characteristic 0. This paper proves that $H$ must possess one of the following two structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\#kG$, where $kG$ is the group algebra of group $G$ of order $p$ and $R$ is a semisimple Yetter–Drinfeld Hopf algebra in $_{kG}^{kG}y\mathcal{D}$ of dimension ${{q}^{n}}$.
This work was partially supported by the NSF of China (11201231), the China Postdoctoral Science Foundation (2012M511643), the Jiangsu Planned Projects for Postdoctoral Research Funds (1102041C), and the Agricultural Machinery Bureau Foundation of Jiangsu Province (GXZ11003).