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I have to do two proofs.

1) If G is abelian, the the factor group G/H is abelian.

2) If H and K are normal in G, then H intersect K is normal in G.

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    These come straight from the definions Susan. Just write them out.2012-12-14

1 Answers 1

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1) Hint: Here you just need to know how you compose elements in $G/H$: $ (gH)(hH) = (gh)H $

2) You want to show that for all $g\in G$, $g(H\cap K)g^{-1}\subseteq H\cap K.$

So let $g\in G$. Now $gNg^{-1} \subseteq N$ and $gKg^{-1} \subseteq K$ because both $H$ and $K$ are normal in $G$. Since $ \begin{align} H\cap K &\subseteq H \\ H\cap K &\subseteq K \end{align} $ we have $ \begin{align} g(H\cap K)g^{-1} &\subseteq gHg^{-1} \subseteq H\quad \text{and}\\ g(H\cap K)g^{-1} &\subseteq gKg^{-1} \subseteq K. \end{align} $ So $g(H\cap K)g^{-1}$ is contained in both $H$ and $K$, hence $ g(H\cap K)g^{-1} \subseteq H\cap K. $