Exceptional isomorphism

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are:[1]

• ${\displaystyle \operatorname {PSL} _{2}(4)\cong \operatorname {PSL} _{2}(5)\cong A_{5},}$ the smallest non-abelian simple group (order 60) – icosahedral symmetry;
• ${\displaystyle \operatorname {PSL} _{2}(7)\cong \operatorname {PSL} _{3}(2),}$ the second-smallest non-abelian simple group (order 168) – PSL(2,7);
• ${\displaystyle \operatorname {PSL} _{2}(9)\cong A_{6};}$
• ${\displaystyle \operatorname {PSL} _{4}(2)\cong A_{8};}$
• ${\displaystyle \operatorname {PSU} _{4}(2)\cong \operatorname {PSp} _{4}(3),}$ between a projective special orthogonal group and a projective symplectic group.

The compound of five tetrahedra expresses the exceptional isomorphism between the icosahedral group and the alternating group on five letters.

There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups:[2]

• ${\displaystyle S_{3}\cong \operatorname {PSL} _{2}(2)\cong {}}$ Dihedral group of order 6,
• ${\displaystyle A_{4}\cong \operatorname {PSL} _{2}(3)\cong {}}$ tetrahedral group,
• ${\displaystyle S_{4}\cong \operatorname {PGL} _{2}(3)\cong \operatorname {PSL} _{2}(\mathbb {Z} /4)\cong {}}$ full tetrahedral group ${\displaystyle \cong }$ octahedral group,
• ${\displaystyle A_{5}\cong \operatorname {PSL} _{2}(4)\cong \operatorname {PSL} _{2}(5)\cong {}}$ icosahedral group,
• ${\displaystyle S_{5}\cong \operatorname {P\Gamma L} _{2}(4)\cong \operatorname {PGL} _{2}(5)}$,
• ${\displaystyle A_{6}\cong \operatorname {PSL} _{2}(9)\cong \operatorname {Sp} _{4}(2)',}$
• ${\displaystyle S_{6}\cong \operatorname {Sp} _{4}(2),}$
• ${\displaystyle A_{8}\cong \operatorname {PSL} _{4}(2)\cong \operatorname {O} _{6}^{+}(2)',}$
• ${\displaystyle S_{8}\cong \operatorname {O} _{6}^{+}(2).}$

These can all be explained in a systematic way by using linear algebra (and the action of ${\displaystyle S_{n}}$ on affine ${\displaystyle n}$-space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for ${\displaystyle A_{8}}$ and ${\displaystyle S_{8}}$ are linked via the exceptional isomorphism ${\displaystyle \operatorname {SL} _{4}/\mu _{2}\cong \operatorname {SO} _{6}}$.)

There are also some coincidences with symmetries of regular polyhedra: the alternating group A₅ agrees with the icosahedral group (itself an exceptional object), and the double cover of the alternating group A₅ is the binary icosahedral group.

The trivial group arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance:

• ${\displaystyle C_{1}}$, the cyclic group of order 1;
• ${\displaystyle A_{0}\cong A_{1}\cong A_{2}}$, the alternating group on 0, 1, or 2 letters;
• ${\displaystyle S_{0}\cong S_{1}}$, the symmetric group on 0 or 1 letters;
• ${\displaystyle \operatorname {GL} (0,\mathbb {K} )\cong \operatorname {SL} (0,\mathbb {K} )\cong \operatorname {PGL} (0,\mathbb {K} )\cong \operatorname {PSL} (0,\mathbb {K} )}$, linear groups of a 0-dimensional vector space;
• ${\displaystyle \operatorname {SL} (1,\mathbb {K} )\cong \operatorname {PGL} (1,\mathbb {K} )\cong \operatorname {PSL} (1,\mathbb {K} )}$, linear groups of a 1-dimensional vector space
• and many others.

The spheres S⁰, S¹, and S³ admit group structures, which can be described in many ways:

• ${\displaystyle S^{0}\cong \operatorname {Spin} (1)\cong \operatorname {O} (1)\cong \mathbb {Z} /2\mathbb {Z} \cong \mathbb {Z} ^{\times }}$, the last being the group of units of the integers;
• ${\displaystyle S^{1}\cong \operatorname {Spin} (2)\cong \operatorname {SO} (2)\cong \operatorname {U} (1)\cong \mathbb {R} /\mathbb {Z} \cong {}}$ circle group;
• ${\displaystyle S^{3}\cong \operatorname {Spin} (3)\cong \operatorname {SU} (2)\cong \operatorname {Sp} (1)\cong {}}$ unit quaternions.

In addition to ${\displaystyle \operatorname {Spin} (1)}$, ${\displaystyle \operatorname {Spin} (2)}$ and ${\displaystyle \operatorname {Spin} (3)}$ above, there are isomorphisms for higher dimensional spin groups:

• ${\displaystyle \operatorname {Spin} (4)\cong \operatorname {Sp} (1)\times \operatorname {Sp} (1)\cong \operatorname {SU} (2)\times \operatorname {SU} (2)}$
• ${\displaystyle \operatorname {Spin} (5)\cong \operatorname {Sp} (2)}$
• ${\displaystyle \operatorname {Spin} (6)\cong \operatorname {SU} (4)}$

Also, Spin(8) has an exceptional order 3 triality automorphism.