Symmetry refers to an attribute of a shape or relation where there is an exact reflection of form on opposite sides of a dividing line or plane. Symmetry occurs not only in geometry but also in other branches of mathematics and can be described as invariance, where something does not change under a set of transformations. Symmetric functions have an output value that remains invariant under permutations of variables, like the example given of (a - b)(b - c)(c - a) = 10. Symmetric matrices and second order partial derivatives of functions are also examples of symmetric functions. Symmetry is related to the mathematical concept of duality.

2. Symmetry means an attribute of a shape or
relation; exact reflection of form on opposite sides
of a dividing line or plane.
Symmetry occurs not only in geometry, but also
in other branches of mathematics. It is actually the
same as invariance: the property that something
does not change under a set of transformations.

3. In the case of symmetric functions, the value of the output is
invariant under permutations of variables. For
example(a − b)(b − c)(c − a) = 10
A symmetric matrix, seen as a symmetric function of the
row- and column number, is an example. The second order
partial derivatives of a suitably smooth function, seen as a
function of the two indexes, is another example.
A high-level concept related to symmetry is mathematical
duality.
A relation is symmetric if and only if the corresponding
boolean-valued function is a symmetric function.

4. In the case of three variables we can use e.g. Schoenflies
notation for symmetries in 3D. In the example the solution
set is geometrically in coordinate space at least of symmetry
type C3. If all permutations were allowed this would be C3v. If
only two unknowns could be interchanged this would be Cs.
In fact, prior to the 20th century, groups were synonymous
with transformation groups (i.e. group actions). It's only
during the early 20th century that the current abstract
definition of a group without any reference to group actions
was used instead.

5. A symmetry of a differential equation is a transformation
that leaves the differential equation invariant, knowledge of
such symmetries may help solve the differential equation.
Symmetries may be found by solving a related set of ordinary
differential equations . Solving these equations is often much
simpler than solving the original differential equations.
Two objects are symmetric to each other with respect to a
given group of operations if one is obtained from the other by
one of the operations. It is an equivalence relation.

6.  The idea of randomness suggests a probability distribution
with "maximum symmetry" with respect to all outcomes.
 A function of two variables is skew-symmetric if f(y, x) =
−f(x, y). The property implies f(x, x) = 0 (except in fields of
characteristic two). A skew-symmetric matrix, seen as a
function of the row- and column number.
 In probability theory, from a symmetry in stochastic events, a
corresponding symmetry of the probability distribution may
be derived. For example, due to the approximate symmetry of
a die each outcome of tossing one, in the sample space {1, 2,
3, 4, 5, 6}, has approximately the same probability.