Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members a_i and b_j of two families (usually infinite) of mathematical objects, that is not an example of a pattern of such isomorphisms. These coincidences are at times considered a matter of trivia, but in other respects they can give rise to other phenomena, notably exceptional objects. In the below, coincidences are listed in all places they occur.

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

In addition to the aforementioned, there are some isomorphisms involving SL, PSL, GL, PGL, and the natural maps between these. For example, the groups over ${\displaystyle \mathbf {F} _{5}}$ have a number of exceptional isomorphisms:

There are coincidences between alternating groups and small groups of Lie type:

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 ${\displaystyle SL_{4}/\mu _{2}\cong SO_{6}}$.) 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.

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