what is the meaning of 2 in group SO(2)?

Question

I am a beginner in group theory. Now reading a note of Lie group. I am confusing about the dimension of the group and the notation. For example, from my understanding, the dimension of SO(2) is 1 because only one parameter (rotation angle) is used to parameterise the group. Then what is the meaning of 2 in the notation of SO(2)? And In the case of SU(2), we have three parameters, correspond to three generators if I understand correctly, then why it is called SU(2)?

Answer 1

You can define the group $SO(2)$ as $2\times 2$ matrices: $$ SO(2) = \left\{\pmatrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)}: \theta\in \mathbb{R}\right\}. $$ So the $2$ comes from the fact that you have $2\times 2$ matrices.

See also here: https://en.wikipedia.org/wiki/Orthogonal_group#Geometric_interpretation

In general $$\begin{align} O(n) &= \{A \in GL(n): A^TA = AA^T = I\}. \\ SO(n) &= \{A \in O(n): det(A) = 1 \}. \end{align} $$ will be a group of $n\times n$ matrices. Note that all the matrices above

• are orthogonal
• have determinant $1$.

Answer 2

One way to understand the definition is to consider quadratic forms. A quadratic form in $\Bbb R^n$ is a real-valued function on $\Bbb R^n$ of the form $$ Q(x_1,...,x_n)=\sum_{i\leq j}a_{i,j}x_ix_j\qquad\qquad(\ast) $$ for some (fixed) coefficients $a_{i,j}\in\Bbb R$ (actually one could give a more intrinsic definition, but this will suffice). It's a theorem due to Sylvester that after a linear change of coordinates, the quadratic form $(\ast)$ can be put in the form $$ Q(X_1,...,X_n)=X_1^2+...+X_p^2-X_{p+1}^2-...-X_{p+q}^2 $$ where $p+q\leq n$. The integers $(p,q)$ are called the signature of $Q$ and depend only on $Q$ and not on the choice of coordinates $(X_i)$.

When $p+q=n$ (i.e. when the form is non-degenerate) one defines ${\rm O}(p,q)$ to be the subgroup of linear transformations of $\Bbb R^n$ that leave $Q$ unchanged, and ${\rm SO}(p,q)$ its subgroups of elements of determinant $1$.

When the signature $(p,q)=(n,0)$ the quadratic form is said definite positive and for short ${\rm O}(n)={\rm O}(n,0)$.