Subgroups of Symmetry Group of Regular Hexagon
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Theorem
Let $\HH = ABCDEF$ be a regular hexagon.
Let $D_6$ denote the symmetry group of $\HH$.
Let $e$ denote the identity mapping
Let $\alpha$ denote rotation of $\HH$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$)
Let $\beta$ denote reflection of $\HH$ in the $AD$ axis.
The subsets of $D_6$ which form its subgroups are as follows:
- $\set e$
- $\set {e, \alpha^3}$
- $\set {e, \beta}$
- $\set {e, \alpha \beta}$
- $\set {e, \alpha^2 \beta}$
- $\set {e, \alpha^3 \beta}$
- $\set {e, \alpha^4 \beta}$
- $\set {e, \alpha^5 \beta}$
- $\set {e, \alpha^2, \alpha^4}$
- $\set {e, \alpha^3, \beta, \alpha^3 \beta}$
- $\set {e, \alpha^3, \alpha \beta, \alpha^4 \beta}$
- $\set {e, \alpha^3, \alpha^2 \beta, \alpha^5 \beta}$
- $\set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$
- $\set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$
- $\set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$
- $D_6$ itself.
Proof
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Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38 \beta$