How many different elements can we obtain by multiplying all element in a group? 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.


The answer to your question is even more subtle. The set of all the possible products is always a coset of the commutator subgroup.
Theorem Let $G$ be a finite group of order $n$, say $G=\{g_1, \dots, g_n\}$ and let $P(G)=\{g_{\sigma(1)}\cdot g_{\sigma(2)} \dots g_{\sigma({n-1})} \cdot g_{\sigma(n)}: \sigma \in S_n\}$. 
(a) If $|G|$ is odd, then $P(G)=G'$
(b) If $|G|$ is even, let $S \in Syl_2(G)$. Then $P(G)=G'$ in case $S$ is non-cyclic. If $S$ is cyclic, then $P(G)=xG'$, where $x$ is the unique element of order $2$ of $S$.
This higly non-trivial and beautiful result relates to combinatorics - the construction of Latin Squares. It heavily relies on the proof of the so-called Hall-Paige conjecture (Hall, Marshall; Paige, L. J. Complete mappings of finite groups. Pacific Journal of Mathematics 5 (1955), no. 4, 541--549.). This could be proved thanks to the classification of the finite simple groups.
