let x be the group z11 under addition and ϕz11 -> z11 be a group decomposed into disjoint cycle notation
ϕ(x)=x^3
show map ϕ is an automorphism of z or z ϵ aut(z11)
Could you rephrase the question exactly as it's written, if I have it wrong? It's a bit difficult to read, at least for me. Here is my interpretation of your question:
For $\displaystyle \varphi : \mathbb{Z}_{11}^{+} \rightarrow \mathbb{Z}_{11}^{+}$, $\displaystyle \forall$ $\displaystyle x \in \mathbb{Z}_{11}^{+}$, $\displaystyle \varphi (x) = x^3$. Prove $\displaystyle \varphi \in \mathfrak{A}|\mathbb{Z}_{11}^{+}|$
Is this what you are asking? If so, could you tell us what it is about this problem that is giving you trouble? I think that the best way to approach this is the same way you would approach any other problem of this sort, where you are attempting to verify whether or not a mapping constitutes a homomorphism/isomorphism. For the 1-1 part, the group is small enough that direct computation is the simplest way to go, although [hint] you can use the fact that 11 is prime to avoid that, by considering the orders of the elements. The fact that this group is abelian also makes your job quite a bit easier.