what is the meaning of 2 in group SO(2)? 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)?
 A: 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 fat that you have $2\times 2$ matrices.
See also here: https://en.wikipedia.org/wiki/Orthogonal_group#Geometric_interpretation
In general
$$\begin{align}
O(n) &= \{A \in GL(n): A^TA = A^TA = I\}. \\
SO(n) &= \{A \in O(n): det(A) = 1 \}.
\end{align}
$$
will be a group of $n\times n$ matrices.
Note that all the matrices above


*

*are orthogonal

*have determinant $1$.

A: 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 ${\rm O}(n)={\rm O}(n,0)$.
