Theorem: Isomorphism between Symmetry Group of Regular Pentagon and Subgroup of Symmetric Group

Let $\PP = ABCDE$ denote a regular pentagon.
Let $\struct {\PP, \circ}$ be the symmetry group of $\PP$, where the various symmetries are identified as:
:the identity mapping $e$
:the rotations $r, r^2, r^3, r^4$ of $72^\circ, 144^\circ, 216^\circ, 288^\circ$ around the center of $\PP$ anticlockwise respectively
:the reflections $t_A, t_B, t_C, t_D, t_E$ about the lines through the center of $\PP$ and the vertices $A$ to $E$ respectively.
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Let $\struct {S_5, \circ}$ denote the symmetry group on $5$ letters.
Then $\PP$ is isomorphic to a subgroup of $\struct {S_5, \circ}$ of order $10$.

Proof:
Let the $A$, $B$, $C$, $D$ and $E$ of $\PP$ be identified with the integers $1$, $2$, $3$, $4$ and $5$ of $S_5$.
Let each of the symmetries of $\PP$ be identified with permutations of $S_n$ according to where the symmetry moves the vertices of $\PP$.
We express these in two-row notation, which we construct by inspection:
{{begin-eqn}}
{{eqn | l = e
      | r = \begin {pmatrix} A & B & C & D & E \\ A & B & C & D & E \end {pmatrix}
      | c = {{Defof|Identity Mapping}}
}}
{{eqn | l = r
      | r = \begin {pmatrix} A & B & C & D & E \\ B & C & D & E & A \end {pmatrix}
      | c = Definition of $r$
}}
{{eqn | l = r^2
      | r = \begin {pmatrix} A & B & C & D & E \\ C & D & E & A & B \end {pmatrix}
      | c = Definition of $r^2$
}}
{{eqn | l = r^3
      | r = \begin {pmatrix} A & B & C & D & E \\ D & E & A & B & C \end {pmatrix}
      | c = Definition of $r^3$
}}
{{eqn | l = r^4
      | r = \begin {pmatrix} A & B & C & D & E \\ E & A & B & C & D \end {pmatrix}
      | c = Definition of $r^4$
}}
{{eqn | l = t_A
      | r = \begin {pmatrix} A & B & C & D & E \\ A & E & D & C & B \end {pmatrix}
      | c = Definition of $t_A$
}}
{{eqn | l = t_B
      | r = \begin {pmatrix} A & B & C & D & E \\ C & B & A & E & D \end {pmatrix}
      | c = Definition of $t_B$
}}
{{eqn | l = t_C
      | r = \begin {pmatrix} A & B & C & D & E \\ E & D & C & B & A \end {pmatrix}
      | c = Definition of $t_C$
}}
{{eqn | l = t_D
      | r = \begin {pmatrix} A & B & C & D & E \\ B & A & E & D & C \end {pmatrix}
      | c = Definition of $t_D$
}}
{{eqn | l = t_E
      | r = \begin {pmatrix} A & B & C & D & E \\ D & C & B & A & E \end {pmatrix}
      | c = Definition of $t_E$
}}
{{end-eqn}}
These are now identified with permutations of $S_5$ and hence elements of $\struct {S_5, \circ}$, which we express in cycle notation:
{{begin-eqn}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ A & B & C & D & E \end {pmatrix}
      | r = \paren 1 \paren 2 \paren 3 \paren 4 \paren 5
      | c = {{Defof|Identity Mapping}}
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ B & C & D & E & A \end {pmatrix}
      | r = \begin {pmatrix} 1 & 2 & 3 & 4 & 5 \end {pmatrix}
      | c = Definition of $r$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ C & D & E & A & B \end {pmatrix}
      | r = \begin {pmatrix} 1 & 3 & 5 & 2 & 4 \end {pmatrix}
      | c = Definition of $r^2$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ D & E & A & B & C \end {pmatrix}
      | r = \begin {pmatrix} 1 & 4 & 2 & 5 & 3 \end {pmatrix}
      | c = Definition of $r^3$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ E & A & B & C & D \end {pmatrix}
      | r = \begin {pmatrix} 1 & 5 & 4 & 3 & 2 \end {pmatrix}
      | c = Definition of $r^4$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ A & E & D & C & B \end {pmatrix}
      | r = \begin {pmatrix} 2 & 5 \end {pmatrix} \begin {pmatrix} 3 & 4 \end {pmatrix}
      | c = Definition of $t_A$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ C & B & A & E & D \end {pmatrix}
      | r = \begin {pmatrix} 1 & 3 \end {pmatrix} \begin {pmatrix} 4 & 5 \end {pmatrix}
      | c = Definition of $t_B$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ E & D & C & B & A \end {pmatrix}
      | r = \begin {pmatrix} 1 & 5 \end {pmatrix} \begin {pmatrix} 2 & 4 \end {pmatrix}
      | c = Definition of $t_C$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ B & A & E & D & C \end {pmatrix}
      | r = \begin {pmatrix} 1 & 2 \end {pmatrix} \begin {pmatrix} 3 & 5 \end {pmatrix}
      | c = Definition of $t_D$
}}
{{eqn | l = \begin {pmatrix} A & B & C & D & E \\ D & C & B & A & E \end {pmatrix}
      | r = \begin {pmatrix} 1 & 4 \end {pmatrix} \begin {pmatrix} 2 & 3 \end {pmatrix}
      | c = Definition of $t_E$
}}
{{end-eqn}}
The isomorphism is implicit in the construction.
{{qed}}