Subgroups of Symmetry Group of Regular Hexagon

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:

Order $1$

$\set e$

Order $2$

$\set {e, \alpha^3}$

$\set {e, \beta}$

$\set {e, \alpha \beta}$

$\set {e, \alpha^2 \beta}$

$\set {e, \alpha^3 \beta}$

$\set {e, \alpha^4 \beta}$

$\set {e, \alpha^5 \beta}$

Order $3$

$\set {e, \alpha^2, \alpha^4}$

Order $4$

$\set {e, \alpha^3, \beta, \alpha^3 \beta}$

$\set {e, \alpha^3, \alpha \beta, \alpha^4 \beta}$

$\set {e, \alpha^3, \alpha^2 \beta, \alpha^5 \beta}$

Order $6$

$\set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$

$\set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$

$\set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$

Order $12$

$D_6$ itself.

Proof

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Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38 \beta$