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**johng** Hi,

Some preliminary observations:

Let $G$ be any group and $A, B$ subsets of $G$. The product set is defined by $AB=\{ab:a\in A \text{ and }b\in B\}$. If A or B is a singleton, write $aB$ or $Ab$ for $AB$. Let $C$ be another subset, easily: if $A\subset B$, then $AC\subset BC$ (also you can multiply by C on the left). So if $A=B$, $AC=BC$.

1. Let $H$ be a subgroup of $G$. Then $H$ is normal in $G$ if and only if $xH=Hx$ for all $x\in G$.

a) Assume $H$ is normal in $G$ and $x\in G$. Then $x^{-1}Hx=H$. Thus $xx^{-1}Hx=xH$ and so $Hx=xH$.

b) Assume $Hx=xH$ for all $x\in G$. Let $x\in G$. Then $x^{-1}Hx=x^{-1}xH=H$. So $H$ is normal in $G$.

2. Assume $H$ is a non-empty subset of G with $Hx=HxH$ for all $x\in G$. Then $xH=Hx$ for all $x\in G$.

I can't work this problem (I may be overlooking something that is obvious). Some comments, though ( 1 is the identity of $G$):

Suppose $1\in H$. Let $x\in G$. Then $1xH\subset HxH=Hx$ and so $xH\subset Hx$. So $H=x^{-1}xH\subset x^{-1}Hx$ for any x. Choosing $x^{-1}$, I get $H\subset (x^{-1})^{-1}Hx^{-1}$ and so $H\subset xHx^{-1}$ and then $x^{-1}Hx\subset H$. Thus $H=x^{-1}Hx$. By the same argument as in 1a), I get $Hx=xH$.

Next $H1=H1H$ and so $H=HH$; i.e $H$ is closed under multiplication. So if $G$ is finite or even if $H$ contains an element of finite order, then $1\in H$ and I'm done. So a counter example must be an infinite non-abelian group with no elements of finite order in $H$. I haven't been able to come up with such an example.

3. Hints: Determine the subgroup $H\cap K$. Next for $h\in H, k\in K$, examine the element $h^{-1}k^{-1}hk$.