Undergrad Group Theory sub algebra of unitary group of U(6) group.

[Vikas Katoch]

TL;DR

three sub algebra of Unitary group (6) as 1. U(5).
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.

three sub algebra of Unitary group (6) as 1. U(5) .
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.

 

[fresh_42]

Summary:: three sub algebra of Unitary group (6) as 1. U(5).
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.

three sub algebra of Unitary group (6) as 1. U(5) .
2. SU(3)
3. O(6)
here the three chains in attachment is attached.
I want to know how these chains are understands in group theory.

Sub groups, not sub algebras. Of course we need to specify each inclusion separately. And it is not really an inclusion in the sense of subsets, they are embeddings in the sense of monomorphisms, injective group homomorphisms.

E.g. $O(n) \hookrightarrow O(n+1)$ can be done by $A\longmapsto \begin{bmatrix}A&0\\0&1\end{bmatrix}$.

 

[Vikas Katoch]

Unitary group of order six U(6) having three sub groups.
How these chains are produced. sheet attached.

 

[fresh_42]

Sorry, I do not download what I don't trust. If you want to type it instead, see
https://www.physicsforums.com/help/latexhelp/

And I bet there are more than three subgroups in $U(6)$.

 

[Vikas Katoch]

Unitary group of order six U(6) having three sub groups.

Type 1. U(6)⊃ U(5) ⊃ O(5)⊃ O(3) ⊃ O(2)

Type 2. U(6) ⊃SU(3) ⊃ O(3) ⊃ O(2)

Type 3. U(6) ⊃O(6) ⊃ O(5) ⊃ O(3) ⊃ O(2)

How these chains are produced.

 

[fresh_42]

Unitary group of order six U(6) having three sub groups.

Type 1. U(6)⊃ U(5) ⊃ O(5)⊃ O(3) ⊃ O(2)

Type 2. U(6) ⊃SU(3) ⊃ O(3) ⊃ O(2)

Type 3. U(6) ⊃O(6) ⊃ O(5) ⊃ O(3) ⊃ O(2)

How these chains are produced.

I already told you in post #2.

 

[Infrared]

I don't think that $O(3)$ is a subgroup of $SU(3)$. The former has elements of determinant $\pm 1$, but $SU(3)$ only has elements of determinant $1$.