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QUESTION


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

\begin{description}

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

\item[(ii)]
If $G$ and $G'$ are isomorphic groups then every subgroup of $G$
is isomorphic to a subgroup of $G'$.

\item[(iii)]
Every group is isomorphic to a subgroup of a permutation group.

\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 a surjective homomorphism then the order
of $G'$ divides the order of $G$.

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

\end{description}



ANSWER


\begin{description}

\item[(i)]
False

If $m,n$ are odd,
$\tau_1,\tau_2,\ldots,\tau_m,\tau_{m+1},\ldots,\tau_{m+n}$ are
transpositions and $\sigma=\tau_1\ldots\tau_m,\
\sigma'=\tau_{m+1}\ldots\tau_{m+n}$ then $\sigma,\sigma'\in\{$ odd
permutations$\}$ but $\sigma\sigma'$ is even.

\item[(ii)]
True

Let $f:G\rightarrow G'$ be an isomorphism, and $H<G$. Then
$f|_H:H\rightarrow f(H)$ is bijective and $\forall h_1,h_2\in H\
f(h_1.h_2)=f(h_1).f(h_2)$ since $f$ is an isomorphism. So $H\cong
f(H)$.

\item[(iii)]
True

Let $S_G$ denote the group of permutations of $G$ and for each
$g\in G$ define $\sigma_g\in S_G$ by $\sigma_g(h)=g.h.\
f:\begin{array}{c}G\Rightarrow S_G\\g\mapsto \sigma_g\end{array}$
is injective by the cancellation lemma and
$f(gg')(h)=\sigma_{gg'}(h)=gg'h=g(g'h)=\sigma_g(g'h)=\sigma_g\circ\sigma_{g'}(h)$
so it is a homomorphism. Hence $f$ is an isomorphism from $G$ to
$Im(f)$.

\item[(iv)]
False

If $n$ is prime then any group of order $n$ is cyclic and hence
abelian.

\item[(v)]
True

Let $K=$ker$f$ so $\frac{G}{K}\cong G'$ and $|G'|=\frac{|G|}{|K|}$
i.e. $|G|=|K||G'|$

\item[(vi)]
True

A subgroup $H<G$ is normal $\Leftrightarrow gH=Hg$ for every $g\in
G$. In an Abelian group $gh=hg\forall h,g\in G$ so $gH=Hg$.

\end{description}




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