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In Aluffi's Algebra: Chapter 0, Chapter 2 has the following exercise:

1.8. Let $G$ be a finite group, with exactly one element $f$ of order $2$. prove that $\prod_{g\in G} g = f$.

I can prove this if $G$ is abelian, but can't seem to prove it for the general case. (I think in the general case, the claim in the question is that the any product of all the group elements (irrespective of the order in which the elements appear in the product) equals $f$.)

On this site, I could find a post for the abelian case.

So, if the question is false, can someone confirm it by giving a counterexample?

Shaun
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    This can't be right: if the product of all elements of a finite group doesn't depend on the order of those elements, one can prove immediately that the group is in fact abelian. The simplest nonabelian group I know with exactly one element of order 2 is $\Bbb Z/2\Bbb Z \times H$ where $H$ is the nonabelian group of order $21$; perhaps there's a counterexample there. – Greg Martin Apr 20 '22 at 00:15
  • @GregMartin Perhaps a stupid question, but: in a nonabelian finite group $G$, can every $p\in G$ be written as a product of each element of $G$ exactly once? – Noah Schweber Apr 20 '22 at 00:49
  • Absent a specific order, the product does not determine a unique element if the group is nonabelian. The linked duplicate shows the assumption that $G$ is abelian is necessary not only for the result to hold, but to even make sense. – Arturo Magidin Apr 20 '22 at 01:50
  • @NoahSchweber: no—in symmetric groups, for example, the parity of the product of all elements is independent of the order. More generally, if $G$ has a normal subgroup $N$ such that $G/N$ is abelian, the product of all elements of $G$ will be constrained by its image in $G/N$. – Greg Martin Apr 20 '22 at 05:51
  • @GregMartin Ah, quite right, I was being silly. – Noah Schweber Apr 20 '22 at 05:53
  • @NoahSchweber It is proved here that you can obtain all elements in the relevant coset of $G'$: https://www.sciencedirect.com/science/article/pii/S0304020808732572 – Sean Eberhard Apr 20 '22 at 08:15

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