(https://en.wikipedia.org/wiki/Abelian_group -- I'll use ⋆ as the Abelian group's operation, and ~ for inversion, below.)
I believe you are implying:
(g(1) ⋆ ... ⋆ g(n)) ⋆ ~(g(i(1)) ⋆ g(i(2)) ⋆ ... ⋆ g(i(n-1))) = g(m)
where "m" is the group element index that is not covered by "i".
However, for this to work, it is requried that you can distribute the inversion ~ over the group operation ⋆, like this:
~(g(i(1)) ⋆ g(i(2)) ⋆ ... ⋆ g(i(n-1))) = ~g(i(1)) ⋆ ~g(i(2)) ⋆ ... ⋆ ~g(i(n-1))
because it is only after this step (i.e., after the distribution) that you can exploit the associativity and commutativity of operation ⋆, and reorder the elements in
g(1) ⋆ ... ⋆ g(n) ⋆ ~g(i(1)) ⋆ ~g(i(2)) ⋆ ... ⋆ ~g(i(n-1))
such that they pairwise cancel out, and leave only the "unmatched" (missing) element -- g(m).
However, where is it stated that inversion ~ can be distributed over group operation ⋆? The above wikipedia article does not spell that out as an axiom.
Wikipedia does mention "antidistributivity":
https://en.wikipedia.org/wiki/Distributive_property#Antidist...
(which does imply the distributivity in question here, once we restore commutativity); however, WP says this property is indeed used as an axiom ("in the more general context of a semigroup with involution"). So why is it not spelled out as one for Abelian groups?
... Does distributivity of inversion ~ over operation ⋆ follow from the other Abelian group axioms / properties? If so, how?