Question:
Let $G$ be a group and $H$ a subgroup of $G$. A conjugacy class of an element $α∈G$ is the set $\def\CC{\mathop{\rm CC}}\CC(a)=\{ g^{-1} ag \mid g∈G\}$. Prove that $H$ is the union of conjugacy classes if and only if $H$ is normal in $G$.
My Answer:
Prove that if $H$ is normal in $G$ then $H$ is the union of conjugacy classes. Assume that $H$ is not the union of the conjugacy classes. This means that $H$ does not have an element that is of the form $g^{-1} ag$ where $a∈G$. Now, $H$ is normal in $G$, then $gH=Hg$ for all $g∈G$. Hence, for some $g∈G$ and $h∈H$ there exist $h'∈H$ such that $gh=h' g$. This shows that $h=g^{-1} h' g∈H$ which contradicts that $H$ does not have any element of the form $g^{-1} ag$, $a∈G$. Thus, it is proven that if $H$ is normal in $G$, then $H$ is the union of conjugacy classes.
Conversely, prove that if $H$ is the union of conjugacy classes then $H$ is normal in $G$. A property of subgroup $N$ being normal to group $M$ is that for all $m∈M$, $mNm^{-1}⊆N$. Must then show that for all $g∈G$, $gHg^{-1}⊆H$. Let $a∈ gHg^{-1}$, then $a=ghg^{-1}$ for some $h∈H$. Now, the element $$a=ghg^{-1}=(g^{-1} )^{-1} hg^{-1}∈H.$$ Thus, by the stated property, $H$ is a normal subgroup in $G$.
Therefore it is proven that $H$ is the union of conjugacy classes if and only if $H$ is normal in $G$.
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