What is a symmetry?
There is no “rigorous definition”. Informally, the notion of symmetry appears wherever we have:
- a collection (set) of transformations (actions, operations, …) which can be applied to these objects and transform them again into objects of the same class;
- a collection (set) of objects (points, figures, variables, functions, rules of the game…).
Then usually, but not always, the result of the transformation of an object is a new, different object. However, there may be exceptions: some objects may stay the same after a nontrivial transformation. Then we say that such objects are symmetric and the transformations preserving the object are called its symmetries.
Sometimes the mere understanding of the symmetry can be a key to solving the problem.
Problem. For which values of the parameters the system of algebraic equations
admits only one solution?
Solution. Consider the transformation of cyclic change of variables, . This transformation can be applied to each of the three equations and preserves all of them. Therefore if we have a solution , then the triple is also a solution. If the system has only one solution, then it should stay the same, i.e., . In other words, all three components should be equal to the same number . The first equation instantly implies that . Substituting this value to the remaining equations, we see that and .
Problem. Can the above system have exactly two solutions for some values of ?
Problem. Consider the regular polygon with vertices, inscribed in a circle. Denote by the center of the circle and the vertices (see the picture for ):
Prove that the vector sum .
Solution. Consider the rigid rotation of the plane around the origin by the angle . Then the circle and the polygon will be preserved, only the labels of the vertices will change as follows: . Thus all terms in the above vector sum will undergo a cyclic permutation, which does not change the result. If we denote this sum by , then the conclusion is that the rotation of by is equal to . Yet the only vector that does not change after a rotation (other than by ) is the zero vector.
Problem. Prove that for integer and prime, the difference is divisible by . (Little Fermat theorem)
Solution. Consider necklaces (מחרוזת) of beads (חרוזים) each, made in distinct colors, and let’s count them. We can choose each bead by distinct ways, so the total number appears to be . However, this is not the right answer, since the necklaces which differ by rotation we counted several times:
A “completely asymmetric” necklace, without any regularity in the pattern, will be counted times, since we have exactly rotations. On the other hand, the monochromatic (single-color) necklaces will be counted only once: there exist of them.
Are there any intermediate symmetric necklaces? If after rotation by “steps” we obtain the initial necklace, then this would mean that the color pattern is periodic with period . But the whole necklace should consist of an entire number of such “periods”, i.e., should be divisible by . But by assumption, is prime! This means, that except for the monochromatic necklaces, there are no “partially symmetric” ones, only completely asymmetric (i.e., such, that any nontrivial rotation produces a different coloring scheme). Now we conclude that
(total count of coloring schemes) = (number of single-color schemes) + (number of non-symmetric colorings, an integer number).
This proves that is divisible by .
Problem. Prove the little Fermat theorem by induction in . Hint: consider the binomial coefficient and show that for prime it is always divisible by except when or .