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Is it an invariance under rotations around x-, y-, z-axis? Does this invariance separately include rotations around an arbitrary $(x$, $y$, $z)$ axis?

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I'd understand this to mean the symmetry group of the (standard, unit) cube. This is generated by rotations by 90 degrees around the standard axes and may or may not mean to include reflections as well (so the group has either $24$ or $48$ elements). – Hagen von Eitzen Dec 18 '12 at 22:37
The rotations around an arbitrary axis through the origin in three real dimensions are the direct symmetries of a sphere. These symmetries form an important continuous group known as $SO(3)$ - adding reflections gives you $O(3)$. The symmetries of a cube are a discrete subgroup (also finite subgroup) of the relevant spherical group (with or without reflections). There is a whole world to investigate arising from this question - including regular polyhedra, and different dimensions. – Mark Bennet Dec 18 '12 at 23:05
So, if I understood you correctly, it's mean, that some tensor for a solid body with cubic symmetry is invariant under rotations around x-, y-, and z-axis. So, it's also invariant under total rotation? – John Taylor Dec 19 '12 at 9:51
up vote 3 down vote accepted

The cubic symmetry group is the group of operations which leave a cube invariant. They are generated by rotations around the $x$, $y$, $z$ axis by $\pi/2$.

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How about reflexions on the $xy$-plane, $xz$-plane and $yz$-plane? – Patrick Da Silva Dec 18 '12 at 22:47
@PatrickDaSilva: you can add them if you like. see here The group with reflections is called $O_h$, without reflections $O$. – Fabian Dec 18 '12 at 22:50
I was asking because I was wondering if "cubic symmetry group" was a standard term for some people. To me it's really just "look at that thing and get a group out of it", so it doesn't feel like reflexions have any reason to be/not to be there... but whatever. =) – Patrick Da Silva Dec 18 '12 at 23:50

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