It is easy to check that we can embed ( injective homomorphism ) $C_3$ in $S_3$, where $C_3$ is a cyclic group of order 3. Just map a order 3 element to order 3 element of $S_3$.

Same idea can be extended to embed $C_n$ into $S_n$, take the generator of $C_n$ and map it into an element of order $n$ in $S_n$.

Question : How many injective homomorphisms are there from $C_n$ to $S_n$?

To me answer seems equal to number of elements of order $n$ in $S_n$

  • 3
    $\begingroup$ Yes, and you can replace $S_n$ by $G$ in the conclusion. $\endgroup$ Feb 20 '18 at 7:50
  • 1
    $\begingroup$ You may interested to know OEIS. $\endgroup$
    – Orat
    Feb 20 '18 at 9:01

Correct. Let $e$ be a generator of $C_n$. If $f\colon C_n\longrightarrow S_n$ is an injective homomorphisme, then $f(e)$ must be an element of order $n$. Since $e$ generates $C_n$, knowing $f(e)$ completely characterizes $f$. Therefore, yes, there are as many such homomorphisms as elements of order $n$ in $S_n$.


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