I want to use Cayley's theorem to determine a subgroup in $S_n$ ( for n as small as possible) which is isomorphic to $C_{5}$.
I believe this subgroup to be $ \langle (1 2 3 4 5) \rangle $. Here is my reasoning:
$C_5 = \{ e, g, g^2, g^3, g^4\}$ is generated and hence determined by $g$. The isomorphism $ \rho$ used in the proof of Cayley's theorem maps $g$ to $(1 2 3 4 5)$ in $S_5$. But since $ \rho $ is a homomorphism, $ \rho(C_5) = \langle (1 2 3 4 5) \rangle.$ Therefore, $C_5 \cong \langle (1 2 3 4 5) \rangle.$
Is this correct? Any help is greatly appreciated!
Best Answer
I don't see what Cayley's theorem has to do with anything; it gives you an embedding of your group into some symmetric group, but gives you no guarantees of that being the smallest symmetric group.
In stead, a quick check will show that $S_n$ has no elements of order $5$ for $n<5$. And you have found a subgroup of order $5$ in $S_5$, hence $n=5$ is the smallest $n$ for which such a subgroup exists. Done.