# Rank of SO(3) and SO(4)?

The rank of SO(3) is 1, the rank of SO(4) is 2. I'm trying to understand the definition of rank of a group with those two examples.

The rank of a group is the cardinality of the smallest generating set. The definition from Wikipedia is given in the first sentence. (Link to wikipedia: https://en.wikipedia.org/wiki/Rank_of_a_group)

Definition of generating set: "a generating set of a group is a subset such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. "

In the case of SO(3), the group operation would be (matrix-)multiplication and there is no way one could express all the uncountably many rotations in the xy-plane with a finite product of matrices.

• It is the rank of its Lie algebra, i.e., the dimension of a Cartan subalgebra - see this post. Commented Mar 27, 2020 at 12:52
• Thank you very much. For my use case (finding the number of Casimir operators) it would be a detour to go over to lie groups anyways. Commented Mar 27, 2020 at 14:21
• Every time anyone defines a function called $f$ which can be applied to both an $X$ and a $Y$, and every $X$ is a $Y$, but the values of $f$ don't agree, Bourbaki kills a kitten. Here, rank is the culprit, but there are many others. On behalf of all of mathematics, I apologize. Commented Mar 27, 2020 at 21:07

Since you're asking about the Lie groups $$SO(3)$$ and $$SO(4)$$, you're looking at the wrong definition of rank.

You don't want the rank of a group meaning the minimal number of generators; for an uncountable group, that rank is uncountable, as you suspected.

Instead you want the rank of a Lie group, and I quote from that link: "For connected compact Lie groups... the rank of the Lie group is the dimension of any one of its maximal tori."

The rank of a Lie-group is the dimension of a maximal torus. In $$SO(3)$$ a maximal torus is given by the rotations around just one axis, for example $$SO(2)\times \{1\} < SO(3)$$, and this is diffeomorphic to $$S^1$$, the circle. Therefore the rank of $$SO(3)$$ is 1 (the dimension of $$S^1$$).

In $$SO(4)$$ you have maximal tori of the form $$SO(2)\times SO(2)$$, so here the maximal torus really is a torus $$S^1 \times S^1$$, which is 2-dimensional. So the rank of $$SO(4)$$ is 2.

Edit: I changed maximal abelian groups to maximal tori. (Thanks for the comment)

• "Torus" is more restrictive than "abelian subgroup": torus is an abelian subgroup consisting of semisimple (diagonalizable) elements. There might be abelian subgroups of bigger dimension than maximal tori, for example $SL(2n)$ whose maximal tori have dimension $2n-1$ has abelian subgroups of dimension $n^2$ (block-triangular matrices $\begin{pmatrix}I_n&A\\0_n&I_n\end{pmatrix}$) Commented Mar 27, 2020 at 21:27