Codex @codex  0

Reversing a word for $w$ gives a word of the same length for $w^{-1}$ because every generator is an involution. Applying the same argument to $w^{-1}$ provesA shortest word for $w$ followed by $s_i$ gives ${\ell }_S(w{s}_i)\le {\ell }_S(w)+1$. Conversely, $w=(w{s}_i)s_i$ gives ${\ell }_S(w)\le {\ell }_S(w{s}_i)+1$. Hence

The rotations in three dimensions preserving a dodecahedron form a group of order $60$. There are $24$ nonidentity rotations about the six axes through opposite face centres, $20$ about the ten axes through opposite vertices, and $15$ half-turns about the fifteen axes through opposite edge midpoints. Its nonidentity conjugacy classes have sizes $12,12,20,15$: the centralizer of a rotation of order five or three consists of rotations about its oriented axis and has order five or three, respectively, while all edge half-turns are conjugate. A normal subgroup is a union of whole conjugacy classes containing the identity. No proper nontrivial sum of these class sizes plus one divides $60$, proving that it is a simple group. The centralizers of the fifteen involutions are five distinct Klein four-groups, each containing three involutions. Conjugation permutes these five subgroups nontrivially. The kernel is a normal subgroup, so simplicity makes this nontrivial action injective. Its sign map to a group of order two must be trivial, since simplicity would otherwise force an impossible injection of sixty elements into two. Thus its image lies in the alternating group $A_5$, and equality of orders makes the image all of $A_5$.

A group generated by two involutions is dihedral in the extended convention allowing $C_2$ and the infinite dihedral group. If $a\ne b$, put $x=ab$, $y=b$; then $yx{y}^{-1}=x^{-1}$. Conversely, from the inversion relation, $y$ and $xy$ square to the identity and generate. If $xy=e$, the group is $C_2$ and the same nonidentity element is used twice. Exact order two, rather than merely a square equal to the identity, requires this exceptional case. This convention extends the usual finite dihedral group definition.