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QUESTION

Decide for each of the following statements whether or not it is
true giving a brief explanation of your answer.

\begin{description}

\item[(i)]
The even permutations in $S_n$ form a subgroup.

\item[(ii)]
The union of any two subgroups of group $G$ is itself a subgroup
of $G$.

\item[(iii)]
Every finite group is isomorphic to a subgroup of $S_n$ for some
positive integer $n$.

\item[(iv)]
For every positive integer $n$ there is a non-abelian group with
precisely $n$ elements.

\item[(v)]
If $f:G\Rightarrow G'$ is an injective homomorphism then the order
of $G$ divides the order of $G'$.

\item[(vi)]
Every subgroup of an abelian group is abelian.

\end{description}



ANSWER


\begin{description}

\item[(i)]
True

$\sigma_i\ i=1,2$ are even $\Leftrightarrow$ each can be written
as products of even numbers of transpositions. The product is also
a product of an even number of transpositions. so is even.

\item[(ii)]
True

Let $H,K<G$. Then for any $h,k\in H\cap K\ hk\in H$ and $hk\in K$
so $hk\in H\cap K$,\ $h^{-1}\in H$ and $h^{-1}\in K$ so $h^{-1}\in
H\cap K$ and $e\in H$ and $e\in K$ so $e\in H\cap K$.

\item[(iii)]
True:

Let $n=|G|$ and choose a bijection $f$ between $G$ and
$\{1,\ldots,n\}$. $f$ induces an isomorphism from $S_G$ to $S_n$.

$G$ embeds in $S_G$ via the left regular representation
$g\mapsto(\sigma_g:h\mapsto gh)$.

\item[(iv)]
False

If $p$ is prime then every group of order $p$ is cyclic and
therefore abelian.

\item[(v)]
True

$f(G)$ is a subgroup of $G'$ so $|f(G)|$ divides $|G'|$.

Since $f$ is injective $|f(G)|+|G|$

\item[(vi)]
True

Let $G$ be abelian, $H<G$. For any $g,h\in G\ gh=hg$. In
particular, for $g,h\in H,\ gh=hg$.

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