Question

Find all closed subgroups 1 ) of the plane \(\boldsymbol{R}^{2} ; 2\) ) of the space \(\boldsymbol{R}^{n} ; 3\) ) of the circle \(S^{1}=\\{z \in C:|z|=1\\}\).

Short answer

Answer: Some examples of closed subgroups in Case 1 (the plane \(\boldsymbol{R}^{2}\)) are lines through the origin and the origin itself. 2. What are some examples of closed subgroups in Case 2? Answer: Some examples of closed subgroups in Case 2 (the vector space \(\boldsymbol{R}^{n}\)) are linear subspaces passing through the origin and the origin itself. 3. What are some examples of closed subgroups in Case 3? Answer: Some examples of closed subgroups in Case 3 (the circle \(S^{1}=\{z \in C : |z|=1\}\)) are sets of points obtained by rotating the unit circle by a fixed angle, such as \(\{1, -1\}\) and \(\{1, e^{i\frac{2\pi}{3}}, e^{i\frac{4\pi}{3}}\}\).

Step-by-step solution

Identify the space

We are working with the plane \(\boldsymbol{R}^{2}\), which is a 2-dimensional space.

Find possible candidate subgroups

Some possible candidates include lines through the origin and the origin itself (as a trivial subgroup).

Check the conditions for closed subgroups

We want to check if these candidate subgroups are closed under the operations of addition and scalar multiplication. Indeed, linear subspaces (which are lines through the origin in \(\boldsymbol{R}^{2}\)) are closed under these operations. Moreover, the origin, being a single point, is also closed under these operations. Therefore, the closed subgroups of the plane are lines through the origin and the origin itself. Case 2: Subgroups of the space \(\boldsymbol{R}^{n}\)

Identify the space

We are working with the vector space \(\boldsymbol{R}^{n}\), which is an n-dimensional Euclidean space.

Find possible candidate subgroups

Some possible candidates include linear subspaces (such as lines, planes, etc.) passing through the origin and the origin itself.

Check the conditions for closed subgroups

All linear subspaces passing through the origin and the origin itself are closed under the operations of vector addition and scalar multiplication. Hence, the closed subgroups of the space \(\boldsymbol{R}^{n}\) are the linear subspaces passing through the origin and the origin itself. Case 3: Subgroups of the circle \(S^{1}=\{z \in C : |z|=1\}\)

Identify the space

We are working with the space \(S^{1}=\{z \in C : |z|=1\}\), which is the unit circle in the complex plane.

Find possible candidate subgroups

Some possible candidates include sets of points obtained by rotating the unit circle by fixed angles, like \(\{1, -1\}\) (corresponding to rotation by \(\pi\)), \(\{1, e^{i\frac{2\pi}{3}}, e^{i\frac{4\pi}{3}}\}\) (corresponding to rotations by \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\)), etc.

Check the conditions for closed subgroups

For a set of points obtained by rotation to be a closed subgroup, it needs to be closed under the operation of complex multiplication. For example, \(\{1, -1\}\) is a closed subgroup because when you multiply any two elements of the set, you still stay within the set. The same is true for the other examples mentioned. Therefore, the closed subgroups of the circle \(S^{1}\) are sets of points obtained by rotating the unit circle by a fixed angle (including the full circle, which corresponds to the trivial subgroup).

Key concepts

Linear Subspaces

A linear subspace is a fundamental idea in linear algebra, which is essentially a collection of vectors that create a 'flat' within a vector space. It's natural to think of it as a line or a plane passing through the origin, depending on the dimensions involved.
In the context of \(oldsymbol{R}^{2}\), when we talk about linear subspaces, we generally refer to lines that go through the origin. What makes these lines special is their closure under two main operations: addition and scalar multiplication. This means if you take any two vectors on the line and add them, or multiply them by a scalar (a non-zero number), the result will still lie on the line.
These properties are why lines through the origin are considered closed subgroups in \(oldsymbol{R}^{2}\). In a more generalized space like \(oldsymbol{R}^{n}\), this concept expands to include higher-dimensional 'flats' that again must pass through the origin and maintain closure under the defined operations. These could be points, lines, planes, or even higher-dimensional analogs. The key takeaway is that linear subspaces are foundational building blocks of vector spaces and maintain consistent structural properties within them.

Complex Plane

The complex plane gives us a geometric representation of complex numbers, merging the real and imaginary parts into a single visual framework. This plane resembles a coordinate system, where every point corresponds to a complex number of the form \(a + bi\), with \(a\) as the real part and \(b\) as the imaginary part.
Within this plane, the concept of closed subgroups gains an interesting twist when viewed through the lens of the unit circle \(S^{1} = \{z \in C : |z|=1\}\). This set represents all numbers that lie exactly at a distance of 1 from the origin of the complex plane, forming a perfect circle. It's not just geometry at play here; this context brings about rotational symmetries that are unique to complex multiplication.
These symmetries allow us to define closed subgroups on the circle, composed of points obtained by rotating the circle by constant angles. Think of it like dividing the circle into equal sections and observing that these sections maintain integrity when you 'spin' them, or multiply their corresponding points together. Thus, sets like \(\{1, e^{i\frac{2\pi}{3}}, e^{i\frac{4\pi}{3}}\}\) form perfect examples of closed subgroups within this space.

Vector Spaces

Vector spaces form the backbone of linear algebra and provide the framework in which linear subspaces exist. Imagine a vector space as a vast "field" where vectors live and operate under defined rules. These rules include addition and scalar multiplication, which are essential for maintaining the structure and operations within the space.
A vector space can be \(oldsymbol{R}^{n}\), \(oldsymbol{C}^{n}\), or any other set equipped with vectors that satisfy specific axioms. Importantly, vector spaces are more than just collections of vectors; they include the operations that govern vector interactions. For instance, any scalable transformation of vectors or interaction of vectors that results in another vector within the same space falls under its canopy.
Linear subspaces are specific "regions" of these vector spaces that satisfy closure requirements. They are critical because they help us understand more complex structures by breaking them down into simpler, manageable parts. Whether we're looking at a line within \(oldsymbol{R}^{2}\) or a plane within \(oldsymbol{R}^{3}\), we're essentially exploring the vast landscape of vector spaces where these simpler forms reside, each having its own set of properties and behaviors.