Automorphisms and Finite Cyclic Groups

Question

The Question: Let C be a finite cylic group and let $ E \subset C $ be a subgroup of C. Prove that every automorphism $\alpha: C\rightarrow C$ we have $\alpha(E) = E$.

I know that every subgroup of a cylic group is cylic, and that for any $e \in E$ that the $o(e) = o(\alpha (e))$ but how can I connect these ideas. Hints would be appreciated thanks.

Well, if you want a hint, how many different subgroups of order $m$ are there in a cyclic group of order $n$? — Matt Samuel, Apr 01 '16 at 01:56
@MattSamuel is it true that in a cyclic group of order N that the order of all its elements is unique. — IntegrateThis, Apr 01 '16 at 02:25

score 1 · Accepted Answer · answered Apr 01 '16 at 20:32

Suppose $E\leq C$ is a subgroup of a finite cyclic group $C$, so $E=\langle g\rangle$ for some $g\in G$. If $\alpha\in\mathrm{Aut}(C)$, then $|\alpha(g)|=|g|$, so $\alpha(E)=\langle \alpha(g)\rangle$ and $$|E|=|g|=|\alpha(g)|=|\alpha(E)|$$ As $C$ has at most one subgroup of a given order, it follows that $E=\alpha(E)$.

Automorphisms and Finite Cyclic Groups

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