# Tag Info

Accepted

### Does SO(3) preserve the cross product?

You may use the scalar triple product formula $r \cdot (p\times q)=\det(r,p,q)$ to prove that $$gr \cdot (gp\times gq)=gr \cdot g(p\times q)\tag{1}$$ ($=\det(r,p,q)$) for any vector $r$. Since $g$ ...
• 123k
Accepted

### Does $G$ being a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for all odd primes $p$ imply it is a subgroup of $GL(n, \mathbb{Z})$?

No, the quaternion group $Q_8$ is a subgroup of ${\rm GL}(2,p)$ for all odd primes $p$, but not of ${\rm GL}(2,{\mathbb Z})$.
• 78.8k
Accepted

### Finite-order elements of $\text{GL}_4(\mathbb{Q})$

Let $A$ be a matrix of finite order $n$. Consider its minimal polynomial $m(x)$. More generally,
• 15.6k
Accepted

• 206k

### Diagonalizability of elements of finite subgroups of general linear group over an algebraically closed field

Let $A\in G$ be a matrix. After changing bases, $A$ is in JNF. Let us write $A=D+N$ with $D$ diagonal and $N$ the nilpotent part. Now since $G$ is finite, there is some $m\in\Bbb N$ with $A^m=I$, the ...
• 14.2k
Accepted

• 33k