The symmetry group of a cube $G$ is commonly cited as being $G \cong S_4 \times \mathbb{Z}_2$. However, the definition of an internal direct product (i.e see this link https://groupprops.subwiki.org/wiki/Internal_direct_product) demands that the elements of $S_4$ i.e our rotations commute with the elements of $\mathbb{Z}_2$ i.e our reflection which isn't true. What should I do? Is there some way to get around this?
2026-03-24 23:46:28.1774395988
Writing the symmetry group of a cube without using direct products
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