The algebraic groups , particularly finite, have significant presence in physics. Both in applications and in research.
The symmetry of a physical system means that its equations of motion remain unchanged with respect to a certain set of transformations.
Let’s highlight the following important property: if an equation is invariant with respect to the hk transformations, it will also be invariant with respect to the l transformation . which is a consequence of the successive application of h and k. The transformation l is called the product of the transformations h and k , so the operation that has just been proposed is an internal operation of the set of symmetry transformations of the given physical system. [one]
Summary
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- 1 group
- 2 Groups used in physics
- 3 References
- 4 Sources
Group
group G is named the set of objects or operations – called group elements – that satisfy the following conditions:
1.- For the set G, a law of internal composition (multiplication) is defined , that is, a law that two arbitrary elements a and b, in a predetermined order, are placed in univocal correspondence with a certain element c of the same G, said element it is called the product of the elements a and b; denoted as c = ab (juxtaposed form of product)
2.- the abstract and generalized operation of this multiplication must verify the associative property ; that is, for arbitrary elements a, b and d of G, the equality a (bd) = (ab) d must be satisfied. the law of composition does not have to fulfill commutativity; in general bd ≠ db. Groups in which two arbitrary elements commute are called abelian groups.
3.- Between the elements of G there is an element e, such that for any element of a of G, we have ae = ea = a. Element e is named unit element of the group.
4.- for every element a of G, there is an element h in G, such that ah = e.
this element is called the inverse element of a and is denoted -1 . It is shown that a single unit element, and that for each element of G, there exists a single inverse element. [two]
Groups used in physics
Group of displacements in three-dimensional space
also called a translation group. Its elements are the translational transformations of the origin of coordinates in an arbitrary vector w , that is, v ‘= v + w’ . It is evident that it is a continuous group.
Rotation group O + (3)
its elements are the rotational transformations of the three-dimensional space or the corresponding orthogonal matrices with a determinant equal to 1. This group is continuous or topological. If the inversion is added to the rotation group: x ‘= -x, y’ = – y, z ‘= – z the orthogonal group O (3) is obtained.
Symmetry groups of molecules
Or specific groups. Its elements are certain orthogonal transformations of three-dimensional space. For that matter, the symmetry group of a molecule in the form of a regular tetrahedron (CH4 methane molecule) consists of 24 elements between rotations and reflections that transform the vertices of the tetrahedron into one another.
Crystal symmetry groups
Groups of permutations of m objects
Lorentz L + Group
