Let $G = \{ g\_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is
$$ G! = [ \prod\_i g\_{\sigma(i)} | \sigma \in S\_n] $$.


I'm interested in knowing how $G!$ behaves as a set, and also how often does every element appear (i.e., how it behaves as a multiset). 


In the case of an abelian $G$, $G!$ is either 1 or (iff exists) the single element of order 2 in $G$. 


We can use the abelian case to get a (seemingly tight) upper bound on $G!$ as a set, by considering modding by $[G,G]$: projecting every such product to the quotient, which is abelian, we get $a^{\#[G,G]} \in G/[G,G]$ where $a$ is the single element of order 2 in $G/[G,G]$ if it exists, and the identity otherwise. Thus, if either $a$ is the identity or $\#[G,G]$ is even, $G!$ is entirely contained in $[G,G]$, and otherwise contained in the coset corresponding to $a$. 


The reason this bound seems tight is that it's generally easy to get elements in the commutator group (up to the element of order 2): if $a,b,a^{-1},b^{-1}$ are distinct, $[a,b] \in G!$ by putting every other element of $G$ right next to it's inverse except for $a^{\pm1}, b^{\pm1}$ and likewise for products of commutators and so on. 


Is it always the case that $G!$ is a coset of the commutator group? How often does each element appear? 
It might also be useful to look at the action of $Aut(G)$ on $G!$, but I'm not totally sure what can that tell us.



Multiplying all the elements in a group
Yes, your $G!$ (as a set) is always either $[G,G]$ (if the order of $G$ is odd, or its Sylow $2$-subgroup is non-cyclic) or $z[G,G]$ if $G$ has cyclic Sylow $2$-subgroup, where $z$ is the involution in the Sylow $2$-subgroup. This was apparently a question/conjecture of [Golomb](https://www.ams.org/journals/bull/1970-76-05/S0002-9904-1970-12518-6/) (see p. 973) and independently of Fuchs (in a 1964 seminar), [proved](https://www.sciencedirect.com/science/article/pii/S0304020808732572) by Dénes and Hermann.


(Addressing Mark Sapir's comment, I could not find a published reference for Fuchs, but did manage to track down [this paper](https://www.sciencedirect.com/science/article/pii/S0195669889800447) by Dénes and Keedwell which contains a discussion of some of the history of the question.)


