Editing Alternating group (section)

== Exceptional isomorphisms ==
There are some [[exceptional isomorphism]]s between some of the small alternating groups and small [[groups of Lie type]], particularly [[projective special linear group]]s. These are:
* A<sub>4</sub> is isomorphic to PSL<sub>2</sub>(3)<ref name="Robinson-p78">Robinson (1996), [{{Google books|plainurl=y|id=lqyCjUFY6WAC|page=78|text=PSL}} p. 78]</ref> and the [[symmetry group]] of chiral [[tetrahedral symmetry]].
* A<sub>5</sub> is isomorphic to PSL<sub>2</sub>(4), PSL<sub>2</sub>(5), and the symmetry group of chiral [[icosahedral symmetry]]. (See<ref name="Robinson-p78"/>  for an indirect isomorphism of {{nowrap|PSL<sub>2</sub>(F<sub>5</sub>) → A<sub>5</sub>}} using a classification of simple groups of order 60, and [[Projective linear group#Action on p points|here]] for a direct proof).
* A<sub>6</sub> is isomorphic to PSL<sub>2</sub>(9) and PSp<sub>4</sub>(2)'.
* A<sub>8</sub> is isomorphic to PSL<sub>4</sub>(2).

More obviously, A<sub>3</sub> is isomorphic to the [[cyclic group]] Z<sub>3</sub>, and A<sub>0</sub>, A<sub>1</sub>, and A<sub>2</sub> are isomorphic to the [[trivial group]] (which is also {{nowrap|1=SL<sub>1</sub>(''q'') = PSL<sub>1</sub>(''q'')}} for any ''q'').
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The associated extensions {{nowrap|SL<sub>''n''(''q'') → PSL<sub>''n''</sub>(''q'')}} are [[universal perfect central extension]]s for A<sub>4</sub>, A<sub>5</sub>, A<sub>8</sub>, by uniqueness of the universal perfect central extension;
for PSL<sub>2</sub>(9) <math>\cong</math> A<sub>6</sub>, the associated extension is a perfect central extension, but not universal: there is a 3-fold [[Schur multiplier|covering group]].
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