# Product of all elements of a finite group of order 2k

I had this problem that said given a finite ring with unity R with 1+1 ≠ 0 such that for a permutation of its invertible elements, their product is equal to 1. It asked to show that 4 divides the order of its invertible elements subgroup.

I was able to show that 2 must divide the order of U(R), but now i’d be curious if given a group of order 2k, where k is odd, is the product of it’s elements always different from 1.

• What is $2$ in your first sentence? Is the question "is it true that if group $G$ has order $2k$ where $k$ is odd, then product of all elements of $G$ in any order is different from $1$"? Commented Nov 13, 2023 at 14:28
• Yes, it's true, and it's a consequence of something called Burnside's transfer theorem. Concretely, consider the regular representation of $G$ on itself, where $g \in G$ acts as $h \mapsto hg$. You can check that an element of order $2$ acts as an odd permutation, and therefore we have a nontrivial homomorphism $G \to C_2$, and the product of all the elements of $G$ (in any order) is again an odd permutation, hence nontrivial. Commented Nov 13, 2023 at 14:33
• Also, what does $1+1\ne 2$ mean? As far as I know $2$ is by definition $1+1$ in any unital ring. Commented Nov 13, 2023 at 14:35
• It was meant to be 0 instead of 2, that was my mistake, oops Commented Nov 13, 2023 at 14:38
• Then isn't $\mathbf Z/3\mathbf Z$ a counterexample? Commented Nov 13, 2023 at 16:57

In addition to the comments above, there is also a very neat answer to the general question $$\color{blue} {\text{What is the set of all different products of all the elements of a finite group G? }}$$ So $$G$$ not necessarily abelian and in a product each element appears exactly one time.

Well, if a $$2$$-Sylow subgroup of $$G$$ is trivial or non-cyclic, then this set equals the commutator subgroup $$G'$$.

If a $$2$$-Sylow subgroup of $$G$$ is (non-trivial) cyclic, then this set is the coset $$xG'$$ of the commutator subgroup, with $$x$$ the unique involution of a $$2$$-Sylow subgroup.

See also J. Dénes and P. Hermann, $$`$$On the product of all elements in a finite group', Ann. Discrete Math. 15 (1982) 105-109. The theorem connects to the theory of Latin Squares and so-called complete maps.