Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true?

Question

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true?

$1.G$ has exactly $4$ subgroups upto isomorphism.

$2.G$ is abelian.

$3.G$ is isomorphiq to a subgroup of $S_{pq}$.

My attempt: First option is true.

Second option is false $S_3$ is a counterexample.

My problem:I think third option is correct.Is it correct or not?

Answer 1

Suppose $G$ has $n$ elements $\{g_1,g_2,\ldots,g_n\}$...

Consider the action $\eta : G\times \{g_1,g_2,\ldots,g_n\}\rightarrow \{g_1,g_2,\ldots,g_n\}$

By this we mean, given $g\in G$ we have $\eta _g : \{g_1,g_2,\ldots,g_n\}\rightarrow \{g_1,g_2,\ldots,g_n\}$ with $g\cdot g_i\mapsto gg_i$

This $\eta_g$ is a permutation...

This $\eta_g$ is a permutation on set of $n$ elements so is an element of $S_n$

So, you have map $\eta: G\rightarrow S_n$ with $g\rightarrow \eta_g$

If you can prove that this is an injective homomorphism then you have $G\leq S_n$

So, any group of order $n$ is a subgroup of symmetric group on $n$ elements...

Can you fill the gaps??

Can you see how this is related to your third question??

Answer 2

The Cayley's Theorem solves your question: every finite group of order $n$ is isomorphic to a subgroup of the permutations group $S_n$.