Yes/No: There always exists an injective homomorphism from $G$ into $S_n$.

Question

Let $G$ be a finite group of order $n\ge2$. Is the following statements true/false?

There always exists an injective homomorphism from $G$ into $S_n$.

My attempt: I found the answer here. I think this statement is False.

Take $G=\mathbb{Z}_{24} $ and $S_8$.

Now $f:\mathbb{Z}_{24} \to S_8$ is not injective.

Here I take $n=8$ and $f$ denotes group homomorphism.

Edit: $f:\mathbb{Z}_{8} \to S_8$ is not injective.

Am I right?

Answer 1

This is actually Cayley's theorem and it is true if $n$ is the order of $G$. The counter-example you suggest has $n = 8 < 24 = |G|$.

Answer 2

The statement is true. You have references in the link you posted.