Is the general linear group compact?

In summary, the general linear group, or GL(n), is the set of all invertible n x n matrices over a given field. It is not compact because it is neither closed nor bounded. An example of a compact subgroup within the general linear group is the special orthogonal group, or SO(n). The general linear group's non-compactness is significant in mathematics as it allows for the study of infinite sets and functions and has implications in fields such as topology and differential geometry.
  • #1
pivoxa15
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Homework Statement


Is the general linear group over the complex numbers compact?

The Attempt at a Solution


I have a feeling it is not. It is not bounded.
 
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  • #2
Correct. It's also not closed.
 

Related to Is the general linear group compact?

1. What is the definition of the general linear group?

The general linear group, denoted as GL(n), is the set of all invertible n x n matrices over a given field. In other words, it is the group of all nonsingular linear transformations on an n-dimensional vector space.

2. Is the general linear group compact?

No, the general linear group is not compact. A group is considered compact if it is both closed and bounded, but the general linear group is neither. This is because the group contains infinitely many elements and does not have a finite limit.

3. Why is the general linear group not compact?

The general linear group is not compact because it is not bounded. As mentioned earlier, the group contains infinitely many elements, meaning that there is no finite limit that the elements of the group approach. Additionally, the group is not closed because it does not contain all of its limit points.

4. What is an example of a compact subgroup of the general linear group?

One example of a compact subgroup of the general linear group is the special orthogonal group, denoted as SO(n). This group consists of all n x n orthogonal matrices with determinant equal to 1. It is a compact subgroup because it is a closed and bounded subset of the general linear group.

5. How is the compactness of the general linear group important in mathematics?

The concept of compactness is important in mathematics because it allows for the study of infinite sets and functions. In the case of the general linear group, its non-compactness has important implications in fields such as topology and differential geometry. The lack of compactness in this group also leads to a deeper understanding of the structure and properties of infinite groups in general.

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