1. Special Linear Group $S{L}_n(\mathbb{R})$

• Definition: $S{L}_n(\mathbb{R})$ is the group of $n\times n$ real matrices with determinant 1.
• Connectedness: For $n\ge 2$, $S{L}_n(\mathbb{R})$ is connected. This means there is a path between any two elements in the group. In fact, it is a simple Lie group, which implies it has no non-trivial normal subgroups.

2. General Linear Group $G{L}_n(\mathbb{R})$

• Definition: $G{L}_n(\mathbb{R})$ is the group of $n\times n$ invertible real matrices.
• Connectedness: $G{L}_n(\mathbb{R})$ is not connected. It has two components:
  • The component consisting of matrices with positive determinant (which is path-connected).
  • The component consisting of matrices with negative determinant (also path-connected). Thus, $G{L}_n(\mathbb{R})$ is a disjoint union of these two components.

3. Special Orthogonal Group $S{O}_n(\mathbb{R})$

• Definition: $S{O}_n(\mathbb{R})$ is the group of $n\times n$ orthogonal matrices with determinant 1.
• Connectedness: $S{O}_n(\mathbb{R})$ is connected for $n\ge 2$. It is a compact Lie group and is path-connected. For $n=2$, it is topologically equivalent to the circle $S^1$.

4. Orthogonal Group $O_n(\mathbb{R})$

• Definition: $O_n(\mathbb{R})$ is the group of $n\times n$ orthogonal matrices with determinant $\pm 1$.
• Connectedness: $O_n(\mathbb{R})$ is not connected. It has two components:
  • The component consisting of matrices with determinant 1, which is $S{O}_n(\mathbb{R})$ (connected).
  • The component consisting of matrices with determinant -1, which is disconnected from $S{O}_n(\mathbb{R})$.

5. Special Unitary Group $S{U}_n(\mathbb{C})$

• Definition: $S{U}_n(\mathbb{C})$ is the group of $n\times n$ unitary matrices with determinant 1.
• Connectedness: $S{U}_n(\mathbb{C})$ is connected for all $n\ge 1$. It is a compact, connected Lie group. For $n=2$, it is topologically equivalent to the 3-sphere $S^3$.

6. Unitary Group $U_n(\mathbb{C})$

• Definition: $U_n(\mathbb{C})$ is the group of $n\times n$ unitary matrices.
• Connectedness: $U_n(\mathbb{C})$ is connected. It is a compact Lie group and has a connected topology. It is the complex generalization of the orthogonal group $O_n(\mathbb{R})$, with all matrices having determinant of absolute value 1, not just $\pm 1$.

Summary

• $S{L}_n(\mathbb{R})$: Connected for $n\ge 2$.
• $G{L}_n(\mathbb{R})$: Not connected; has two components based on the sign of the determinant.
• $S{O}_n(\mathbb{R})$: Connected for $n\ge 2$.
• $O_n(\mathbb{R})$: Not connected; has two components.
• $S{U}_n(\mathbb{C})$: Connected for all $n\ge 1$.
• $U_n(\mathbb{C})$: Connected.