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Published online by Cambridge University Press: 28 November 2018
Let 
$n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of 
$(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set 
$\mathbb{Z}/n\mathbb{Z}$. In particular, let 
$p$ be an odd prime and let 
$\unicode[STIX]{x1D6FC}$ be a positive integer. If 
$H_{k}$ is a subgroup of 
$(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index 
$k=p^{\unicode[STIX]{x1D6FD}}u$ such that 
$0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and 
$u\mid p-1$, then where 
$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$
$\unicode[STIX]{x1D711}(n)$ is the Euler totient function.
This work was partially supported by the Grant No. 11471162 from NNSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20133207110012).
