Identify \[ \operatorname{Aut}(\ZZ/p\ZZ), \qquad \operatorname{Aut}((\ZZ/p\ZZ)^n), \qquad \operatorname{Aut}(\ZZ/n\ZZ). \]
Solution
For the additive cyclic group \(C_p=\ZZ/p\ZZ\), every automorphism is determined by the image of \(1\), which may be any nonzero residue class. Thus \[ \operatorname{Aut}(\ZZ/p\ZZ)\cong(\ZZ/p\ZZ)^\times. \] Since \(p\) is prime, this is a cyclic group of order \(p-1\).
The group \[ (\ZZ/p\ZZ)^n \] is the additive group of the \(n\)-dimensional vector space \(\FF_p^n\). Its group automorphisms are exactly the invertible \(\FF_p\)-linear maps, so \[ \operatorname{Aut}((\ZZ/p\ZZ)^n)\cong GL_n(\FF_p). \]
Finally, for the additive cyclic group \(\ZZ/n\ZZ\), an endomorphism is determined by \[ 1\longmapsto a\pmod n. \] It is invertible exactly when \(a\) is a unit modulo \(n\). Therefore \[ \operatorname{Aut}(\ZZ/n\ZZ)\cong(\ZZ/n\ZZ)^\times. \]