We know that if $ \scriptstyle\phi(g)\,\,=\,\,g^{\prime} $, then $ \scriptstyle\phi^{-1}(g^{\prime})\,\,=\,\,\{x\in G\,|\,\phi(x)\,=\,g^{\prime}\}\,\,=\,\,g\,ker(\phi) $.
So, if $ \scriptstyle\phi\,:\,U(30)\to U(30)\,=\,\{1,7,11,13,17,19,23,29\} $, $ \scriptstyle ker(\phi)\,\,=\,\,\{1,11\} $, and $ \scriptstyle\phi(7)\,\,=\,\,7 $, then:
$ \scriptstyle\phi^{-1}(7)\,\,=\,\,7\,ker(\phi)\,\,=\,\,\{7*1\,mod\,30,7*11\,mod\,30\}\,\,=\,\,\{7,17\} $.
- --Nick Rupley 23:58, 1 October 2008 (UTC)
