About the symmetric sum

I want to know when is the symmetric equals some integer times a cyclic sum, in other words $$\sum_{sym}f(a,b,c)= n\cdot\sum_{cyc}f(a,b,c)$$ I realized that this happens quite often, for example $$\sum_{sym} a^4=2(a^4+b^4+c^4)= 2\cdot\sum_{cyc}a^4 $$ or $$\sum_{sym} abc= 2\cdot\sum_{cyc}abc$$

We probably should consider a function with $n$ variables, but I think $2$ or $3$ is enough.

A simple group-theoretical approach brings an natural explanation/condition:

• The LHS summation is over all $\sigma \in S_3$ (group with 6 elements),

• The RHS summation is over a subgroup of $S_3$ named $A_3:=\{Id, \gamma, \gamma^2\}$ with 3 elements, where $\gamma$ is the cycle $a \to b \to c \to a$ ($A_3$ is called the "alternating group").

See the group table here for example.

Therefore, you will generate all $S_3$ if you add to subgroup $A_3$ a permutation, either $a \leftrightarrow b$, or $b \leftrightarrow c$ or $c \leftrightarrow a$.

Therefore, a necessary and sufficient condition for your relationship to hold is that your function $f$ is invariant through one of the permutations of $S_3$,

This is the case for the example $f(a,b,c):=abc$ you give.

A counterexample: $f(a,b,c):=a+\frac{b}{c}$.

Coefficient $2$ is plainly explained by the ratio $\frac63$ of groups' orders (the ratio of "sizes" of $S_3$ vs. $A_3$).

This can be extended to more than 3 variables.