## 10 Jan Applications of Sylow Theorems

Sylow Theorem's is Group Theory are an important mathematical tool. Below we give a simple application of the theorems.

Theorem: If $H$ and $K$ are subgroups of $G$ and $Hleq N_G(K)$, then $HK$ is a subgroup of $G$. In particular, if $K$ is normal is $G$ then $HKleq G$ for any $Hleq G$.

Proof: We prove $HK=KH$. We let $hin H, k in K$.
By assumption, $hkh^{-1}in K$, hence $hk=(hkh^{-1})h in KH$.
This proves that $HKleq KH$.
Similarly, $kh=h(h^{-1}kh)in HK$, proving the reverse.
We know that if $H$ and $K$ are subgroups of a group then $HK$ is a subgroup if and if only $HK=KH$.
This proves the result.

Theorem: Let $G$ be a group of order $pq$, where $p$ and $q$ are primes such that $p.

• If $pmid(q-1)$, there exists a non-abelian group of order $pq$.
• Any two non-abelian groups of order $pq$ are isomorphic.

Proof: We let $Q$ be a Sylow $q$-subgroup of the symmetric group of degree $q$, $S_q$. We know that if $p$ is a prime and $P$ is a subgroup of $S_p$ of order $p$, then $mid N_{S_p}(P) mid = p(p-1)$.
We know that every conjugate of $P$ contains exactly $p-1$ $p$-cycles and computing the index of $N_{S_p}(P)$ in $S_p$ we can prove the above result.
So, $mid N_{S_q}(Q) mid =q(q-1)$.
Since $p mid q-1$ and by Cauchy's theorem $N_{S_q}(Q)$ has a subgroup $P$ of order $p$.
Using the previous theorem we can see that $PQ$ is a group of order $pq$.
Since $C_{S_q}(Q)=Q$ so, $PQ$ is non-abelian.
This proves the first result.

Let $G$ be any group of order $pq$, let $Pin Syl_p(G)$ and let $Qin Syl_q(G)$.
We have $pmid q-1$ and let $p=$.
Since $Aut(Q)$ is cyclic, it contains a unique subgroup of order $p$, say $$, and any homomorphism $phi : P rightarrow Aut(Q)$ must map $y$ to a power of $gamma$.
There are therefore $p$ homomorphisms $phi_i:P rightarrow Aut(Q)$ given by $phi_i(y)=gamma^i, 0leq i leq p-1$.
Now each $phi_i$ for $i$ not equal to  give rise to a non-abelian group $G_i$, of order $pq$.
It is straightforward to check that these groups are all isomorphic because for each $phi_i,~i>0$, there is some generator $y_i$ of $P$ such that $phi_i(y_i)=gamma$.
Thus up to a choice for the arbitary generator of $P$, these semi-direct products are all the same.