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QUESTION
\begin{description}
\item[(a)]
Say what it means for a subset $H$ of a group $(G,e,*)$ to be a
subgroup.
\item[(b)]
Giving brief reasons for your answer, say how many elements of
$S_4$ lie in a subgroup generated by the two elements (12) and
(234).
\item[(c)]
Show that a non-empty subset $H$ of a group $(G,e,*)$ is a
subgroup if $g*h^{-1}\in H$ for all $g,h\in H$.
\item[(d)]
Say what it means for a subgroup to be normal. Show that the
kernel of a homomorphism is always a normal subgroup, first using
the result stated in part (c) to show that it is a subgroup.
\item[(e)]
State Lagrange's theorem and use it to show that any group of
prime order must be cyclic.
\end{description}
ANSWER
\begin{description}
\item[(a)]
A subgroup of a group $(G,e,*)$ is a subset $H\subseteq G$
satisfying the following conditions.
\begin{description}
\item[S1)]
If $h,k\in H$, then $h*k\in H$.
\item[S2)]
The identity element of $e\in G$ is also an element of $H$.
\item[S3)]
If $h\in H$ then $h^{-1}\in H$.
\end{description}
\item[(b)]
They all do, since conjugating (12) by the powers of (234) yields
the transpositions (12),(13),(14) which generate all of $S_4$.
\item[(c)]
Since $H$ is non empty we can choose an element $h\in H$. Putting
$g=h$ we see that $e=g*h^{-1}=h*h^{-1}$ is an element of $H$ by
the hypothesis, so $H$ satisfies axiom S2. Now given any element
$h\in H$ we can put $g=e$ to get $h^{-1}=e*h^{-1}$ as an element
of $H$, so $H$ satisfies axiom S3. Finally given any two elements
$g,h\in h$ we see that $g,h^{-1}\in H$ so $g*(h^{-1})^{-1}\in H$,
or $g*h\in H$. So $H$ also satisfies axiom S1.
\item[(d)]
A subgroup $H$ as required.
\end{description}
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