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**Homework Statement**

Show that [itex]U(55)^3 = \{x^3 \, : \, x \in U(55)\}[/itex] is U(55).

**The attempt at a solution**

I wrote a Perl script that computes both U(55) and [itex]U(55)^3[/itex] and they both are equal. However, I want to solve this algebraically. My attempt is to define [itex]f(x) = x^3[/itex] (mod 55) and demonstrate that f is an automorphism. If f is an automorphism, surely U(55) = [itex]U(55)^3[/itex]. I'm having trouble showing that f is injective.

Suppose f(a) = f(b) and [itex]a \ne b[/itex] where a, b are in U(55). f(a) = f(b) is equiv. to [itex]a^3 = b^3[/itex] (mod 55). I need is reduce this latter equation to a = b (mod 55) but I'm unable to do so.