Equivalence of definitions of reducible multiary group

Statement
Recall that a multiary group is a $$n$$-ary group for some $$n \ge 2$$. The following are equivalent for a $$n$$-ary group with multiplication $$f:G^n \to G$$:


 * 1) There exists a group structure on $$G$$ such that $$f(a_1,a_2,\dots,a_n) = a_1a_2\dots a_n$$ for all (possibly repeated) $$a_1,a_2,\dot,a_n \in G$$, with the multiplication on the right being as per the group structure. For more on this, see group is n-ary group for all n
 * 2) There exists a neutral element $$e \in G$$ for $$f$$: $$f$$ evaluated at any tuple where $$n - 1$$ of the entries are equal to $$e$$ and the remaining entry is $$a \in G$$, gives output $$a$$. This is true regardless of where we place $$a$$ and is also true if $$a = e$$.

Related facts

 * Characterization of subgroup of neutral elements of reducible multiary group
 * Groups giving same reducible multiary group are isomorphic

(1) implies (2)
This is immediate: we can take $$e$$ to be the neutral element (i.e., the identity element) of $$G$$ as a group.

(2) implies (1)
Given: A set $$G$$, function $$f:G^n \to G$$ making it a $$n$$-ary group, an element $$e$$ that is neutral for $$f$$.

To prove: There exists a group structure on $$G$$ with respect to which $$f(a_1,a_2,\dots,a_n) = a_1a_2 \dots a_n$$ for all $$a_1,a_2,\dots,a_n \in G$$.

Proof: We use the notation $$e_i$$ to denote $$e$$ written in succession $$i$$ times.

Caution about non-uniqueness of neutral elements
For a reducible multiary group, there may be many different choices of neutral element. Each choice yields a potentially different binary operation, though the group structures we obtain are isomorphic. For instance, for the $$n$$-ary group obtained from the cyclic group of order $$n - 1$$, every element is a neutral element. The different group structures correspond to affine shifts in the original group, i.e., relabeling all elements by adding a group element to them.