For the sake of the argument, let me work on two specific examples in order to make sense, mathematically speaking, of what exactly we are going to consider as a symmetry of something. The first one: reflection.
Let's fix an axis, as the picture shows. In doing so, we can see that there will be a symmetry of the hole picture, namely a reflection through the such an axis. In other words, in applying a reflection (don't forget the axis), it seems we have done nothing to the picture, then such a reflexion is going to be considered as a symmetry of our picture. It turns out that we may think of that reflection as an element of a set which has indeed an algebraic structure: it is a "group".
Needless to say that in the latter example, once we have fixed the axes, there is only one symmetry of the picture. This means that the "group" of symmetries has two elements: doing nothing(it is certainly a symmetry) and the unique reflection; a group of two elements then.
Another basics example is going to be rotations.
The ancient Greek certainly knew that there are only 5 regular solid in the space. Such solid with clear symmetries we would like to describe in mathematical terms. We are focus on the symmetries of one of them: Tetrahedron.
What do we mean by a symmetry in this context? Well, similarly as Garay pointed out "post", if we make the tetrahedron to spin 120 degrees (fixing an axis), it'll seem that we have done nothing to such a solid. In other words, such type of rotations seem to do nothing to the tetrahedron, that's exactly what we are going to consider as a symmetry of such a solid, ergo rotations of this type (apparently doing nothing) are symmetries in this context.
It turns out the set of rotation of the tetrahedron has algebraic structure. By Algebraic structure we mean, they form a group: A_4, the alternating group in four elements. Therefore, let me say that the set of symmetries of such a solid is the group A_4. Here is a picture showing such symmetries.
In working on the previous examples we have avoided the name of "group action". However it is exactly what we have been doing above. A group is acting on an object by symmetries. The group with two elements was the first example while the alternating group A_4 was the second one. Now we want to mention another less obvious (Csar's opinion) example of symmetry. In doing so and in the light of the previous examples, we might say that we are about to describe another group action and that is exactly the point. It is the point because via that idea, we're going to be able to write the concept of symmetry in a concrete mathematical way. That fact enable us to deal with symmetries abstractly, and such a thing will draw and amazing conclusion working out one of the most famous problems of the last century: Solubility of a polynomial by radicals.
The 18th-century question we are going to deal with is the following. Given a polynomial, when there exists an algebraic expression for its solutions involving only taking radicals, addition and multiplication of its coefficients?. When? and what are the polynomials like whose solutions are a basic algebraic manipulation of its coefficients? The answer to this question was an achievement of the concept of symmetry in terms of group actions solved by a 22-years-old guy named Evariste Galois.
Let's state the theorem exactly as it is:
(Galois-1830) A polynomial with coefficients in a field of characteristic zero is solvable by radicals if and only if its Galois group is solvable.
We are going to argue that given a polynomial, there exists a group acting on its roots by permutations. For example, let us consider the polynomial X^3-1. This polynomial factors as (X-1)(X^2+X+1). Clearly 1 is a root. However there are 2 complex roots as well, and those prevent the "symmetries" of this polynomial from being more than 2. Indeed, we will have 2 symmetries here, namely only permutation the roots of X^2+X+1. In other words, if A and B are roots of the latter polynomial then what we can only do is exchange A by B and vice versa. Therefore, we can say that the associated group of symmetries here is the one with two elements: Z_2. We had seen this group in the first example. Permutations that's all. This has nothing to do with multiply them nor add them up. Just permutations. Since we can't permute a real root for a complex one, that's the reason we can't permute the root 1 neither with A nor B.
It turns out that given a polynomial E. Galois realized there is always a group acting on the set of its roots. The subtle thing, though, is that not all the permutations are allowed. Indeed, those permutation that the symmetries do allow to come into the game, form the so-called Galois group of the the polynomial. Hence, the problem now is to find out whether a permutation is allowed or not. The theorem says "If the symmetries (permutations) of the polynomial are such that they form a group which is both acting on the set of roots and it is solvable, then the polynomial is solvable". In other words, Galois realized that if the permutations acting on the roots of the polynomial form a solvable group then there should exists an algebraic expression for such roots in terms of the coefficients. Moreover, such an algebraic expression should only involve radicals, addition, subtraction and division, i.e., basic algebraic manipulations. He proved that and wrote down his name in the mathematical history in such a beautiful way.
p.s Did you realize that there are no pictures once we started talking about algebra?
That's how it is...no pictures. Just formal mental manipulations.
References
Algebra, Serge Lang. 3rd edition, Springer-Verlag 2002.
Field and Galois Theory, P. Morandi. Springer-Verlag 1996.
Que simetrias tiene la Ritonga?
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