Let $G$ be a finite group.
How many different elements can we obtain by multiplying all element in a group?
Of course, if $G$ is abelian the answer is one but when G is non-abelian, changing the order of the multiplication may produce new elements.
My second question is actually related to my attempt to solve the first one.
Let $S$ be set of all elements produced by multiplying all elements in $G$. Then, it is easy to show that $Aut(G)$ acts on $S$ naturally. I wonder whether this can be transitive.