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QUESTION


Define the following terms:

\begin{description}

\item[(i)]
subgroup

\item[(ii)]
left coset

\item[(iii)]
the order of an element $g$ in a group $G$.

\end{description}

State Lagrange's theorem and use it to show that if $H$ and $K$
are finite subgroups of a group $G$ with $|H|$ and $|K|$ coprime,
then $H\cap K=\{e\}$.

Show that, if in addition $H$ and $K$ are both normal subgroups of
$G$. then for any elements $h\in H$ and $k\in K$, the product
$h^{-1}k^{-1}hk=e$.

Give an example of a finite group $G$ which contains a non-normal
subgroup $H$.




ANSWER


\begin{description}

\item[(i)]
Let $(G,e,*)$ be a group. A subgroup of $(G,e,*)$ is a subset
$H<F$ such that

\begin{description}

\item[(s1)]
$e\in H$

\item[(s2)]
If $h_1,h_2\in H$ then $h_18h_2\in H$

\item[(s3)]
If $h\in H$ then $h^{-1}\in H$.

\end{description}

\item[(ii)]
if $H$ is a subgroup of $G$ and $g\in G$ then the left coset $gH$
is the subset $\{gh|h\in H\}$

\item[(iii)]
The order of $g$ is the least prime integer $n$ such that $g^n=e$,
or is $\infty$ if there is no such $n$.

\end{description}

Lagrange's Theorem

Let $G$ be a finite group and $H<G$ then $|H|$ divides $|G|$.

$H\cap K$ is a subgroup of $H$ so $|H\cap K|$ divides $|H|$.
Similarly it divides $|K|$. Since their greatest common divisor is
1, $|H\cap K|=1$.

The element $h^{-1}k^{-1}h\in K$ since $K\triangle G$ so
$h^{-1}k^{-1}hk\in K$. Similarly $h^{-1}k^{-1}hk\in H$ so
$h^{-1}k^{-1}hk\in H\cap K=\{e\}$.

Let $G=D_n,\ H=\left<\sigma_i\right>\
\rho^{-1}H\rho=\left<\sigma_{i+1}\right>=\left<\sigma_i\right>$ or
similar. $\rho$=rotation and $\sigma_i$=reflection.




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