Symmetry Group is a Group
Theorem
Let $P$ be a geometric figure.
Let $S_P$ be the set of all symmetries of $P$.
Let $\circ$ denote composition of mappings.
The symmetry group $\left({S_P, \circ}\right)$ is indeed a group.
Proof
As a symmetry is a bijection, we can use the result Group of Permutations is a Group.
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Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.3$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (5)$
- John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.7$
