Group of symmetries of a parallelepiped in three dimension.

Question

Describe the group of symmetries of a parallelepiped $-a\leq x\leq a$, $-b\leq y\leq b$, $-c\leq z\leq c$ in $\mathbb{R}^3$ with $a\neq b\neq c.$

I know that a dihedral group is the group of symmetries of a regular polygon. But in our problem, the parallelepiped is not a regular polygon. How to think about this group?

Answer 1

It is going to depend very heavily on what you mean by symmetries.

It is probably meaning either actions induced by GL(n,F) on the space in which the parallelepiped is embedded which fix the parallelepiped in some way, or restricting the actions further by looking at actions induced by some subgroup of GL(n,F) which fix the parallelepiped.

You are probably looking at linear isometries in 3d which act invariantly on the parallelepiped. So moving along as if that is what you mean by symmetries, your group will have the 3 generators

$f(x,y,z) = (-x,y,z)$

$f(x,y,z) = (x,-y,z)$

$f(x,y,z) = (x,y,-z)$

There would be more maps in the case $a=b$, $b=c$, or $a=c$.

The reason I chose isometries as the maps you were probably interested in is that you took the time to specify $a\neq b \neq c$.

This rules out rotations along with changes of coordinate.

So the group is $Z_{2} + Z_{2} + Z_{2}$. Sorry for the notation I'm still getting used to math jax.