Symmetric or skew-symmetric function condition needs to be checked only on a generating set for the symmetric group

Statement
Suppose $$A$$ is a set and $$B$$ is a group. Suppose $$n$$ is a natural number and $$f:A^n \to B$$ is a function. Then:


 * 1) To check that $$f$$ is completely symmetric in all variables (i.e., it is symmetric in every pair of variables) it suffices to check, for a generating set of $$S_n$$, that $$f$$ is invariant under the action of that generating set. More generally, to check that $$f$$ is invariant under a particular subgroup of $$S_n$$, it suffices to check invariance under a generating set for that subgroup.
 * 2) To check that $$f$$ is alternating in all variables (i.e., it is alternating in every pair of variables) it suffices to check, for a generating set of $$S_n$$, that the effect of that permutation on the $$f$$-value coincides with the sign of $$f$$.
 * 3) More generally, for any subgroup $$H$$ of $$S_n$$ and any homomorphism $$\varphi:H \to \operatorname{Aut}(B)$$, the verification that $$f \circ \sigma^{-1} = \varphi(\sigma) \circ f \ \forall \ \sigma \in H$$ is equivalent to the same check done only on a generating set.

Facts used

 * 1) uses::Homomorphism is determined by value on generating set