MHB Symmetry in the algebraic expressions

[NotaMathPerson]

Hello!

I read about the symmetry of the following product

$(x+a+b)(x+b+c)(x+c+a)$ my book says that a, b and c occur symmetrically. Why is that?

 

[MarkFL]

Hello!

I read about the symmetry of the following product

$(x+a+b)(x+b+c)(x+c+a)$ my book says that a, b and c occur symmetrically. Why is that?

This is also called "cyclic symmetry" and it simply means that if you switch the two variables in any of the 3 pairs in $(a,b,c)$, you still have the same expression. Try it to verify. :)

 

[Deveno]

Actually, it's "fully symmetric" which is *different* than cyclic symmetry. To see the difference, consider:

$(a - b)(b - c)(c - a)$

If we send $a \to b \to c \to a$, we get:

$(b - c)(c - a)(a - b)$, which is the same expression (even though the factors are in a different order).

If we just switch $a$ and $b$, we get

$(b - a)(a - c)(c - b) = -(a - b)(b - c)(c - a)$, which is *not* the original expression, but its negative.

That is-if we can swap any pair and get the original expression, it is fully symmetric (permutations are generated by pair-swaps), full symmetry implies cyclic symmetry, but the reverse is not so.

(Note that $(a - b)(b - c)(c - a)$ also remains unchanged under the transformation $a \to c \to b \to a$ which gives:

$(c - a)(a - b)(b - c)$).

The transformation (substitution) $a \to b \to c \to a$ is called a 3-cycle, as it changes 3 things to something else, and repeating it three times "completes the cycle" and leaves you where you were.