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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)]
For each positive integer $n\geq2$ the symmetric group $S_n$ has a
subgroup of index 2.

\item[(ii)]
The function $f:D_n\longrightarrow Z_2$ defined by $f(g)=1$ if and
only if $g$ is a rotation (The set of rotations includes the
identity) and $f(g)=0$ if and only if $g$ is a reflection is a
homomorphism.

\item[(iii)]
There are precisely 48 elements in the cyclic group $Z_{180}$ with
the property that they each generate the whole group.

\item[(iv)]
Given any finite group $G$ there is a positive integer $n$ such
that $G$ is isomorphic to a subgroup of $S_n$.

\item[(v)]
Every group of even order is abelian.

\item[(vi)]
If $G$ is a finite group of order $n$ then $g^n=e$ for every
element $g\in G$.

\end{description}




ANSWER


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\item[(i)]
True, $A_n<S_n$ has index 2

\item[(ii)]
False, $f(\rho^2)=1$ but $f(\rho)+f(\rho)=0$

\item[(iii)]
True, Number of generators of $Z_{180}$ is $\phi(18)).\
180=2^2.5.9$ so $\phi(180)=\phi(2^2).\phi(5).\phi(9)=1.4.6=48$

\item[(iv)]
True, Cayley's theorem gives an isomorphism from $G$ to a subgroup
of $S_g$ and thus into $S_{|G|}$.

\item[(v)]
False, $D_3$ is not abelian but has order 6

\item[(vi)]
True, By Lagrange's theorem the order $d$ of $g$ divides $n$ so
$g^n=(g^d)^{\frac{n}{d}}=e$.

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