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QUESTION
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\item[(a)]
Define the following terms
\begin{description}
\item[(i)]
direct product,
\item[(ii)]
isomorphism,
\item[(iii)]
normal subgroup.
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\item[(b)]
Show that the kernel of a homomorphism is a normal subgroup (you
may assume that it is a subgroup).
\item[(c)]
Let $G$ be a group with identity element $e$ and let $H$ and $K$
be subgroups of $G$ with $H\cap K=\{e\}$. Show that if $hk=kh$ for
any $h\in H$ and any $k\in K$ then the function $f:H\times
K\longrightarrow G$ given $f(h,k)=hk$ is an injective
homomorphism. Show that if $G$ is a group in which every element
has order 2 then $G$ is abelian, and deduce that any two
non-identity elements of $G$ generate a subgroup isomorphic to the
Klein 4-group.
Give an example to show that an abelian group can contain two
elements of order 3 without containing a subgroup isomorphic to
$Z_3\times Z_3$.
\end{description}
ANSWER
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\item[(a)]
\begin{description}
\item[(i)]
$(G,*),(H,.)$ are groups.
$\{(g,h)|g\in G,h\in H\}=G\times H$ with
$(g_1,h_1)(g_2,h_2)=(g_1*g_2,h_1.h_2)$ is the direct product.
\item[(ii)]
An isomorphism is a bijective function $f:G\longrightarrow H$ with
$f(g*k)=f(g).f(k)\forall g,k\in G$.
\item[(iii)]
A subgroup $H=\left\times\left$ since
the map $f:\left\longrightarrow G$ is an
isomorphism onto its image.
$C_3$ contains 2 elements of order 2 but is not isomorphic to
$c_3\times C_3$
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