Stabilizer groups
Introduction
In the field of mathematics, particularly in group theory and its applications, stabilizer groups play a crucial role. A stabilizer group, also known as an isotropy group or little group, is a subgroup of a given group that leaves a particular element of a set invariant under the group action. This concept is fundamental in understanding the symmetry properties of various mathematical structures and has applications in areas such as geometry, algebra, and physics.
Definition and Basic Properties
A stabilizer group is defined in the context of a group action. Let \( G \) be a group acting on a set \( X \). For an element \( x \in X \), the stabilizer of \( x \) in \( G \), denoted \( G_x \), is the set of all elements in \( G \) that fix \( x \): \[ G_x = \{ g \in G \mid g \cdot x = x \}. \]
The stabilizer group \( G_x \) is a subgroup of \( G \). This follows from the fact that the identity element of \( G \) fixes every element of \( X \), and the set of elements fixing \( x \) is closed under the group operation and taking inverses.
Examples
Permutation Groups
Consider the symmetric group \( S_n \), which consists of all permutations of the set \(\{1, 2, \ldots, n\}\). For a fixed element \( i \in \{1, 2, \ldots, n\} \), the stabilizer of \( i \) in \( S_n \), denoted \( (S_n)_i \), is the subgroup of permutations that leave \( i \) unchanged. This stabilizer group is isomorphic to the symmetric group \( S_{n-1} \), which acts on the remaining \( n-1 \) elements.
Linear Algebra
In linear algebra, consider the general linear group \( GL(n, \mathbb{R}) \), which consists of all invertible \( n \times n \) matrices over the real numbers. The stabilizer of a vector \( v \in \mathbb{R}^n \) under the action of \( GL(n, \mathbb{R}) \) is the set of all matrices that map \( v \) to itself. This stabilizer group is isomorphic to the group of invertible matrices of the form: \[ \begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix}, \] where \( A \) is an invertible \((n-1) \times (n-1)\) matrix.
Orbit-Stabilizer Theorem
The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of the orbit of an element to the size of its stabilizer group. Let \( G \) be a finite group acting on a set \( X \), and let \( x \in X \). The orbit of \( x \), denoted \( G \cdot x \), is the set of elements in \( X \) that can be reached by applying elements of \( G \) to \( x \): \[ G \cdot x = \{ g \cdot x \mid g \in G \}. \]
The orbit-stabilizer theorem states that the size of the orbit of \( x \) is equal to the index of the stabilizer of \( x \) in \( G \): \[ |G \cdot x| = [G : G_x]. \]
This theorem provides a powerful tool for counting and analyzing the structure of group actions.
Applications
Geometry
In geometry, stabilizer groups are used to study the symmetries of geometric objects. For example, the stabilizer of a point on a sphere under the action of the rotation group \( SO(3) \) is the subgroup of rotations that fix that point, which is isomorphic to the rotation group \( SO(2) \) of the plane perpendicular to the point.
Physics
In physics, stabilizer groups are important in the study of symmetries and conservation laws. For instance, in the theory of relativity, the stabilizer group of a point in spacetime under the Lorentz group is the group of Lorentz transformations that leave the point invariant. This group is related to the symmetries of physical laws in different reference frames.
Further Concepts
Normalizers and Centralizers
The concept of stabilizer groups is closely related to the notions of normalizers and centralizers. The normalizer of a subgroup \( H \) in a group \( G \), denoted \( N_G(H) \), is the set of elements in \( G \) that conjugate \( H \) to itself: \[ N_G(H) = \{ g \in G \mid gHg^{-1} = H \}. \]
The centralizer of an element \( x \in G \), denoted \( C_G(x) \), is the set of elements in \( G \) that commute with \( x \): \[ C_G(x) = \{ g \in G \mid gx = xg \}. \]
These concepts are useful in understanding the structure of stabilizer groups and their role in group actions.