Set of Rotations is Subgroup of Symmetry Group

Theorem

Let $G$ be a symmetry group.

Let $H$ be the subset of $G$ consisting of the rotations in $G$ about a given axis.

Then $H$ is a subgroup of $G$.

Proof

This theorem requires a proof. In particular: Needs a more formal definition of rotation. Surprised this hasn't already been covered properly. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Example $4.5$