Group Generated by 45-Degree Rotations?

Question

What is the group generated by 45-degree rotations about the (I, J, K) axes in three-dimensional space? Is it a finite group?

Answer 1

Interesting question. The generators are $$A=\pmatrix{1/\sqrt2&1/\sqrt2&0\\-1/\sqrt2&1/\sqrt2&0\\0&0&1},$$ $$B=\pmatrix{1/\sqrt2&0&1/\sqrt2\\0&1&0\\-1/\sqrt2&0&1/\sqrt2}$$ and $$C=\pmatrix{1&0&0\\0&1/\sqrt2&1/\sqrt2\\0&-1/\sqrt2&1/\sqrt2}.$$ Then $$AB=\pmatrix{1/2&1/\sqrt2&1/2\\-1/2&1\sqrt2&-1/2\\-1\sqrt2&0&1/\sqrt2}.$$ $AB$ has trace $1/2+\sqrt2$. This is not an algebraic integer. The trace of any orthogonal matrix of finite order is the sum of roots of unity and is an algebraic integer. So $AB$ has infinite order and therefore $A$ and $B$ generate an infinite group.